First Year Calculus
Course Topic:
Limits
Video Link:
http://oyc.yale.edu/astronomy/astr160/lecture10
Time: 10:30  12:25
University: Yale
Course: Frontiers and Controversies in
Astrophysics
Professor Name: Charles Bailyn
Teaching Ideas: This video shows is uses some basic limits
to come up with three major results in relativity including the famous E = mc^{2}.
This can be used when first introducing limits to show why they are needed to
understand profound ideas of our universe. The student needs to know very little
about physics or calculus to understand this clip.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture14
Time: 62:52 to 64:37
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the momentum in the
context of Einstein's special relativity and shows that the limit of momentum
goes to infinity as the speed of the particle goes to the speed of light.
The professor does not say or write down the word "limit" but he does talk about
getting close to the speed of light resulting in large momentum. This can
be used when talking about limits not existing but going to infinity.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture19
Time: 13:52 to 15:03
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video demonstrates that light must be
massless. The professor writes down the Einstein's equations and states
that since the denominator goes to 0, the numerator must also in order to have a
limit. Thus the mass is equal to 0. This is a great example of
limits where the expression is of indeterminate form.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture3/
Time: 3:07  3:45
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video starts with the equation that
gives the force between an electron and a nucleus as a function of radius.
The professor goes over the limits as the radius goes to infinity and goes to 0.
This is a very simple example of infinite limits, horizontal and vertical
asymptotes.
Course Topic:
Slope and the Derivative
Video Link:
http://oyc.yale.edu/chemistry/chem125a/lecture2
Time: 11:49  16:20
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video shows an application of the
derivative to find how far apart bonded atoms need to get before they break
their bonds. The professor shows that the force is the slope (derivative)
of the energy. He has an animated diagram that shows the parabola and
moving slope lines for Hooke's Law vs. a similar diagram for electrical charge
forces. Then he shows what happens if there are two such forces which
demonstrates that derivatives are additive. He next shows what a minimum
looks like for the sum of the two energies. This is a helpful vides to
show how the graphical version of calculus is used in chemistry.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture31
Time: 11:43 to 15:39
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video looks at the average rate of a
chemical reaction and the instantaneous rate of a chemical reaction. The
context is for chemistry, but the explanation could easily have come right from
a first quarter calculus class. This is a fantastic reinforcement to what
is done on the first day of explaining what a derivative is.
Course Topic:
Derivatives
Video Link:
http://oyc.yale.edu/economics/econ251/lecture21
Time: 67:01 to 67:49
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video looks at the rate of change in
the present value of a bond with respect to the interest rate. The present
value formula is given as a geometric series and the professor calculates the
derivate of each term. It is a simple example of the power rule with a
negative exponent applied multiple times.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture1
Time: 33:28 to 36:30
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video justifies the definition of
velocity as the derivative of the position function. The professor goes through
the full derivation starting with the average velocity and then taking the limit
as Dt goes to 0. He even discusses the
tangent line and its slope. This will be an excellent reinforcement of the
first day of the derivative.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture1
Time: 55:44 to58:04
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows the standard question, "How
high does it go?" that is asked about positionvelocityacceleration for an
object that is moving with constant acceleration.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 18:14 to 18:58
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video starts out with the work
equation DK = Fd and divides both sides by
Dt to get dK/dt = Fv which the professor
describes as the Power. This is an example of taking the derivative of
both sides of an equation and getting a new important physics property.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture13
Time: 25:25 to 28:47
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the relativistic
formula that compares the velocity an object from two frames of reference.
It shows that velocity is relative to the frames and depends on the speed of
light. The professor uses the definition of the derivative (Δx/Δt)
as both go to zero. This is a powerful example of the definition of the
derivative in use and can be shown on the first day of derivatives.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture22
Time: 65:52 to 69:12
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
specific heat at constant volume. The professor takes a derivation of the
very simple function 3/2 RT with respect to T. This can be shown at the
very beginning of the lecture on derivatives.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture12/
Time: 22:20 to 24:28
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video states that the derivative of the
charge is the current and then explains that since the capacitance is assumed to
be constant, it can be pulled out of the derivative. This is an important
application to the constant multiple rule for derivatives.
Video Link:
Video Link: http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture20/
Time: 41:31  48:44 (Skip the middle five minutes or
so to save time)
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video explains how to build a
"differentiator box" which is a circuit that starts out with a voltage v and
produces a voltage dv/dt. This is a cute application of derivatives in
circuit analysis. The professor does not actually do any derivatives, but
instead shows how to create a physical differentiator.
Course Topic:
Higher Order Derivatives
Video Link:
http://oyc.yale.edu/physics/phys200/lecture1
Time: 36:30 to 37:23
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains that once you know how
to take a first derivative one can take the derivative any number of times.
The professor emphasizes that the second derivate is the acceleration while
after that the rest are not that useful. This is a very easy to understand
commentary on higher order derivative that would be helpful to calculus students
to see that the exact same concepts are discussed in physics classes.
Video Link:
http://ocw.mit.edu/courses/chemistry/5112principlesofchemicalsciencefall2005/videolectures/lecture5matterasawave/
Time: 42:50 to 45:41
University: MIT
Course: Principles of Chemical Science
Professor Name: Sylvia Ceyer
Teaching Ideas: This video investigates the wave equation
solution of Schrodinger's and finds it's second derivative. The result is
that the second derivative is a constant multiple of the original function.
This uses the chain rule, but the professor does not show the steps.
Student's can be asked to fill in the details.
Course Topic:
Derivative of a Piecewise Linear Function
Video Link:
http://oyc.yale.edu/economics/econ251/lecture2
Time: 39:22 to 42:36
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video presents the marginal utility
function based on a person who values one ticket more than two tickets and
values three the least. The professor draws the graph of the piecewise
linear function and then marginal utility function that is the derivative graph.
This can be used to explain why the derivative of a continuous function may not
be continuous.
Course Topic:
Tangent Lines
Video Link:
http://oyc.yale.edu/economics/econ25211/lecture8
Time: 21:37 to 26:40
University: Yale
Course: Financial Markets
Professor Name: Robert J Shiller (Nobel Laureate)
Teaching Ideas: This video looks at the scenario where there
are two people on an island who both grow grain. The first consumes a lot
the first year and the second is a saver and consumes a lot the second year.
The production and indifference curves are tangent to each other for each.
They meet up and work out a loan to create a better economy for both. The
professor uses tangent lines multiple times in order to solve this problem.
It is shown completely graphically with no equations presented, but students
will see a clear application of tangent lines. They will need to be told
what a utility curve is in economics in order to understand and they may need to
hear a little about how loans work. The background information is
presented before this clip, so an instructor who is not familiar with economic
theory will want to watch that part first so that the a brief explanation can be
provided to the students.
Video Link:
https://www.youtube.com/watch?v=2H6mltgONi8&index=36&list=PL8A25592E6D32C753
Time: 30:39 to 33:12
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video hints at the way that neural
networks calculate the correct weights. The professor shows that at each
point if the tangent line is not horizontal then the next try is in the downward
direction of the tangent line. The process is continued until the minimum
is reached. This is a unique application of the tangent line that will
interest students.
Course Topic:
Product Rule
Video Link:
http://theoreticalminimum.com/courses/generalrelativity/2012/fall/lecture9
Time: 107:00  107:30
University: Stanford
Course: General Relativity
Professor Name: Leonard Susskind
Teaching Ideas: This video uses the product rule to derive
part of the Einstein Field Equations in general relativity. It relates the
Einstein Tensor to the Ricci Tensor and the Curvature Scalar. This will be
way over the heads of all of the students, but it is good to show them what will
be coming in the future if they become physicists and want to understand that
Einstein's general theory of relativity makes extensive use of calculus.
Video Link:
https://www.youtube.com/watch?v=cVbB6wFNqYc&index=4&list=PLE73AA240E8655D16
Time: 26:43  29:22
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video explains that the commutator
satisfies Leibnitz' Rule. This provides additional reinforcement of the
way the product rule for differentiation works.
Course Topic:
Chain Rule
Video Link: http://oyc.yale.edu/chemistry/chem125b/lecture20
Time: 23:00  27:00
University: Yale
Course: Freshman Organic Chemistry II
Professor Name: Michael McBride
Teaching Ideas: This video uses Hook's Law and Newton's F =
ma to derive the fact that the frequency will be independent of the amplitude.
At the end, the professor shows how this can be used to make a watch. The
chain rule is used in taking the derivative of x = hsin(wt) with respect to t.
This is an easy and elegant application of the chain rule.
Video Link:
http://oyc.yale.edu/economics/econ251/lecture23
Time: 20:58 to 21:59
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video explains mathematically why there
is less risk to diversify rather than put all your money into one stock.
This is based on taking a derivative that uses the chain rule and noticing that
the derivative is negative at zero hence the variance goes down by diversifying.
The differentiation is not that difficult, but the chain rule is used.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 61:00 to 63:03
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video starts with simple harmonic
motion and takes two derivatives using the chain rule each time. The
professor states that the velocity's amplitude is multiplied by ω and the
acceleration is multiplied by ω^{2}. This emphasizes that
the derivative of the inside is what makes velocity and acceleration differ from
each other and from position.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture11
Time: 25:54 to 28:01
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains how a generator works.
In the mathematical derivation, the chain rule is used to find the derivative of
cos(ωt) with respect to t. This is a very simple use of the chain
rule, but helps students understand a fundamental engineering principle.
Course Topic:
Relative Extrema
Video Link:
http://oyc.yale.edu/economics/econ25211/lecture15#transcript
Time: 48:00 to 49:45
University: Yale
Course: Financial Markets
Professor Name: Robert J Shiller (Nobel Laureate)
Teaching Ideas: This video displays the futures curve for
oil. There is a clear maximum and minimum. Of note is that in
hindsight the futures market predicted the oil prices four years out almost
perfectly. This can be used to show an application of relative maximum and
minimum and increasing and decreasing functions. There is no explicit
calculus used, but it is not difficult to infer how a model of the curve could
be constructed and then analyzed.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture16
Time: 24:43 to 28:03
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses optimization to find the
point along a mirror that light will hit in order for it to get to the other
given point. This proves that the angle of incidence equals the angle of
reflection. This is a tough be standard application of extrema problems.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture16
Time: 28:33 to 33:00
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses optimization to find derive
Snell's Law. The professor relates it to the lifeguard problem. The
professor works out the equations and takes a derivative and sets it equal to
zero, but does not show the complicated steps of using the chain rule. An
instructor may need to pause at the point where the derivative is taken and
explain the steps or have the students work out the derivative.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture7/
Time: 6:01  9:31
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video looks at the distance an electron
is from the nucleus. The graph of the radial probability distribution is
shown and the professor explains that when we talk about how far the electron is
from the nucleus, we are only talking about the distance that is of maximum
probability. The equation is not shown, but this clip demonstrates an
important application of relative extrema.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture7/
Time: 33:58  35:54
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video displays the radian probability
functions for the 3s, 3p, and 3d orbitals. The 3s and 3p orbitals have
multiple relative maximums and the focus of this clip is to look at the relative
maximum that is closest to the nucleus (smallest value of r). This is a
nice example that shows that sometimes we are more concerned with a relative
maximum that is not the global maximum.
Video Link:
http://ocw.mit.edu/courses/chemistry/5112principlesofchemicalsciencefall2005/videolectures/lecture16intermolecularinteractions/
Time: 39:15 to 42:49
University: MIT
Course: Principles of Chemical Science
Professor Name: Sylvia Ceyer
Teaching Ideas: This video graphs and explains the Leonard
Jones Potential equation. This is a 12th order polynomial, but has just
two terms. The professor explains how to but does not actually work out
the details of the calculus involved in finding the radius at which the minimum
potential occurs. This would be a great exercise for calculus students to
do.
Video Link:
https://www.youtube.com/watch?v=tBn5gVXr0R4&list=PL8A25592E6D32C753&index=6
Time: 33:13 to 34:36
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video explains that in the hill
climbing search algorithm, one may get stuck at the local maximum instead of the
goal of the local maximum. The professor does this by picture rather than
equations so it is easy to follow and establish intuition about the difference
between local and global extrema.
Course Topic:
Second Derivative Test, Concavity and Inflection
Points
Video Link: http://oyc.yale.edu/economics/econ159/lecture6
Time: 30:45 to 33:17
University: Yale
Course: Game Theory
Professor Name: Ben Polak
Teaching Ideas: This video uses the first and second
derivative tests to find the optimal strategy using game theory. The
calculus is explicitly shown and after a second derivative is taken there is a
clear maximum. At 8:04 the professor describes the problem. A couple
wants to meet up at the movies. There are three movies out: A guy
movie, a chick flick, and Snow White. The couple forgot to tell each other
their plan and they have no means of communicating with each other. They
really want to meet up (first priority), but also would rather not see the
opposite gender movie (lesser priority). The math solves the game theory
strategy of what probabilities the man and woman should choose each of the three
movies.
Video Link:
http://oyc.yale.edu/astronomy/astr160/lecture20
Time: 38:00  40:08
University: Yale
Course: Frontiers and Controversies in
Astrophysics
Professor Name: Charles Bailyn
Teaching Ideas: This video shows an application of concavity
to the future of the universe. The professor shows that the red shift
observed demonstrates that the universe's expansion is accelerating which
implies Dark Energy. The professor does not explicitly use the word
"concave up" but if the students learn that F = ma = mx'', then they will see
that the second derivative greater than 0 implies positive acceleration and
positive concavity. The Dark Energy is often called "Einstein's Biggest
Mistake" due to the fact that when Einstein first came up with the idea he
thought is must be wrong and thus a mistake. Thus Einstein had it correct
all along. The video does not state this, but this the the "+C" in the
integration in Einstein's calculations.
Video Link:
http://oyc.yale.edu/chemistry/chem125a/lecture2
Time: 19:00  20:25
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video shows an application of the
inflection point to finding when chemical bonds are broken. The professor
shows a nice animation of the Morse Potential which measures the bond energy vs.
the distance the atoms are apart from each other. He show that for three
atoms, the potential is additive and at the inflection point, the point at which
the curvature changes from "being this way to this way" as the professor shows
with his hands. The students should at this point be asked what math words
could have been used and the students should respond "concave up" and "concave
down".
Video Link: http://oyc.yale.edu/chemistry/chem125a/lecture7
Time: 47:43  49:17
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video shows both the graphs of the
function and its second derivative on the same xyplane to demonstrate the
relationship between the Psi function from Schrodinger's equation and the
potential energy function. The professor uses "curvature" instead of
"concavity", so that will have to be explained to the students. The graph
is busy but clear. The fives minutes of lecture before this explains how
the second derivative graph is found geometrically.
Video Link:
http://oyc.yale.edu/economics/econ25211/lecture11
Time: 21:46 to 24:16
University: Yale
Course: Financial Markets
Professor Name: Robert J Shiller (Nobel Laureate)
Teaching Ideas: This video presents the "Value Function"
which models prospect theory which shows the way people value financial gains
and losses. For a positive gain the function is concave down and for
losses, the function is concave down. The professor draws a typical graph,
but does not show the equations. It is not differentiable at the origin.
This is because there is a big psychological difference between losing and
gaining even if it is not a significant loss. For the next couple of
minutes, the professor explains it and how businesses use it to exploit people.
This is a nice application of concavity that will interest the students.
Video Link:
http://oyc.yale.edu/economics/econ251/lecture11
Time: 14:20 to 15:55
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video shows the graph of the average
wage in a career vs. the percent of the average wage that a person gets from
Social Security at retirement. The professor specifically points out that
the function is concave showing that Social Security is a better deal for low
wage earners. This is a very practical use of the second derivative that
just about everyone understands.
Video Link:
http://oyc.yale.edu/economics/econ159/lecture4
Time: 38:16 to 43:46 (or 45:02 to see it finally solved)
University: Yale
Course: Game Theory
Professor Name: Ben Polak
Teaching Ideas: This video uses the definition of what is
means to be a "best strategy" in game theory from the prior clip with a specific
example. The professor takes a derivative and then takes a second
derivative noticing that the second derivative is negative so the point is a
maximum. The explanation of the premise comes before, but it takes several
minutes. It involves a synergistic profit sharing agreement. An
instructor may want to just give a brief explanation of what is going on to the
students including writing down the equations for the students. This is a
very well explained use of the first and second derivative test.
Video Link:
https://www.youtube.com/watch?v=K1sqYL0dVqg&list=PL0A0E275BC354C934&index=7
Time: 15:36 to 16:37
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video looks at the slope of hillsides.
The professor says that you want a hill that goes from concave to straight to
convex. An instructor can explain that this is the same as concave up to
an inflection point to concave down. Next the professor humorously relates
a bad slope to acne where the slope switches back and forth several times.
The professor does not do any math, but does show a graph. An instructor
can have the students analyze the graph for concavity.
Video Link:
https://www.youtube.com/watch?v=IgaTSenVDwI&list=PL48DE756A5800ED5F&index=5
Time: 13:52 to 16:43
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video explains the harvest model that
follows the logistics growth model. The professor shows the curve and
explains that the inflection point is the point of maximum growth. Then
she shows the graph of the derivative and explain that this point is the maximum
of the derivative curve. This is a great graphical display of the
inflection point and how to graph the derivative given the function's graph.
Video Link:
https://www.youtube.com/watch?v=_pe60buCnxE&index=20&list=PL6MuV0DF6AuoviA41dtji6qPM4hvAcNk
Time: 44:27 to 47:03
University: UC Berkeley
Course: Artificial Intelligence
Professor Name: Nick Hay
Teaching Ideas: This video goes over the maximum likelihood
method to decide what probability to set for an even given a training set.
The professor writes down the equation and describes how the calculus would be
used to find the maximum likelihood and then use the second derivative test to
determine if it is a max. The instructor can ask students to verify the
result that the professor is given by working out the derivative calculations.
Course Topic:
Newton's Method
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6006introductiontoalgorithmsfall2011/lecturevideos/lecture11integerarithmetickaratsubamultiplication/
Time: 22:53 to 27:44
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video goes over Newton's method and
then shows that it has quadratic convergence so that the number of iterations
needed for n digit precision is logarithmic. This is above and beyond what
is learned in the standard calculus class, but it is an important point in any
algorithm.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6006introductiontoalgorithmsfall2011/lecturevideos/lecture12squarerootsnewtonsmethod/
Time: 4:07 to 8:53
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video proves that Newton's method has quadratic convergence
at least for the square root function. This is a great extension to what
is taught in a calculus class on Newton's method.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6006introductiontoalgorithmsfall2011/lecturevideos/lecture12squarerootsnewtonsmethod/
Time: 26:20 to30:16
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video uses Newton's method to turn
division into multiplication which is built into computers. It is a clear
and relevant application of Newton's method. If you go until 35:32, you
can see an example done.
Course Topic:
Differentials
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 56:54 to 59:43
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains the concept of
differentials and gives the example F(x) = x^{2}. This is almost
exactly what is shown in the corresponding section in calculus class, so rather
an an application it will show that the concept is so important it is explained
again in physics. This is particularly necessary since most students
consider this a minor topic that can be safely forgotten. If you watch
until 62:50, you will see the example F(x) = (1 + x)^{n}.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 63:23 to 65:01
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses differentials to
approximate the relativistic mass that is typically done in a modern physics
class. It makes use of differentials, but the instructor may have to fill
in a couple of steps so that the students can see the connection.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture7/
Time: 37:59 41:08
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses differentials to approximate
a small change in current given a small change in a signal. The circuit language
is pretty high level, but the calculus is described exactly as one would in a
calculus class. If you watch until 44:59, the professor explains
graphically exactly as is done in a calculus class on differentials.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture10/
Time: 44:10 to 46:44
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video goes through the mathematics to
show that a presented electric circuit is a linear amplifier. The
professor does this by using differentials without using the word
"differentials". The engineering application will not be evident to the
students from looking at this clip, so the instructor will need to explain that
this is all about linear amplifiers. The larger A is, the greater the
amplification. Amplifiers are not just used to make music louder.
Their main application is for digital signals so that a receiver can better
distinguish between 0s and 1s. This is how computers get information
through the Internet.
Course Topic:
Integrals and Sums
Video Link:
http://oyc.yale.edu/economics/econ25211/lecture2
Time: 12:35 to 16:15
University: Yale
Course: Financial Markets
Professor Name: Robert J Shiller (Nobel Laureate)
Teaching Ideas: This video looks at the mean and expected
value defined by sums and integrals. This is a nice example that can be
shown to calculus students when they are first learning about integration to
emphasize that integration is much more than just an area under a curve.
The professor only gives and explains the definitions. No examples are
provided here. An instructor can make up examples such as the uniform
distribution to show students how this works.
Course Topic:
Fundamental Theorem of Calculus
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 23:32 to 24:39
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video presents the statement of
the First Fundamental Theorem of Calculus in a clear manner. The proof is
not given, but the statement is written down just as one would get in a calculus
class. This would be a nice reinforcement to present right after the proof
is completed in class. If you go until 26:22, the professor gives a simple
example.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture15part1/
Time: 15:02  18:51
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video presents the equations for
circuits: i = cdv/dt and v = Ldi/dt. The professor integrates both sides
of the second equation to get the integral equation: 1/L int(vdt) = i.
This is a quick and easy application of integrating to get rid of the
derivative. Then he uses this equation to solve a circuit. The last
step is to take a derivative of both sides which uses the second fundamental
theorem of calculus. This is a practical example of the second fundamental
theorem of calculus.
Course Topic:
Basic Integration
Video Link:
http://oyc.yale.edu/physics/phys200/lecture7
Time: 56:24 to 57:12
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the antiderivative of 1/x2
in order to find the gravitational potential energy. The professor calls
it "Mickey Mouse calculus" because it is so easy. The instructor can let
students know that the power rule is used so often in calculus and physics that
soon they will feel like it is "Mickey Mouse calculus".
Video Link:
http://oyc.yale.edu/physics/phys201/lecture1
Time: 58:30 to 62:59
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the force on a charge
that is raised above a charged ring. Although there are quite a few messy
constants, the actual integrand is just a constant. This can be shown to
students who are first learning about integration.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture5
Time: 5:42 to 7:52
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the Force Energy Theorem
using basic integration. The derivation also uses implicit differentiation
backwards which will be a nice review of past material. This is an easy to
follow video that employs simple calculus to arrive at an important physics
theorem. If you watch the video until 10:05, the full conservation of
energy law is derived.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture7
Time: 12:31 to 14:44
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video goes over the force and energy on
a charge when there are two charges and an infinite plate between them. The
integral is so basic, the professor doesn't even write down the integral sign.
This is a quick and simple example of an application of electricity at requires
integration.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture9
Time: 32:37 to 32:59
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
magnet field along the zaxis through a circular wire that is perpendicular to
it. The integrand looks like a tough challenge until the professor points
out that the entire integrand is just a constant. One idea is to pause the
video at the very beginning of the clip and see if any of them see that it is
just a constant. This will help to emphasize that no matter how messy an
integral looks, if the variable of integration is not present, then it is just
the constant times that variable.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture11
Time: 60:27 to 61:56
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the power
of an LR circuit. This involves using implicit differentiation backwards
to write the integrand as a derivative of a function and then just cancelling
the integral and the derivative. This is a nice use of the fact that the
integral really is just the antiderivative.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture3/
Time: 3:45  6:17
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video shows the differential equation
from F = ma and Coulomb's Law that predicts that if there that were the whole
story, the electron would plummet into the nucleus in a small fraction of a
nanosecond. The professor does not go through the calculus, but the
students can be asked to try to derive it themselves. They only need to
know how to integrate using the power rule.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6034artificialintelligencefall2010/lecturevideos/lecture2reasoninggoaltreesandproblemsolving/
Time: 41:08  42:23
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video discusses how much knowledge is
used by a computer in order to solve every integration problems that could be
given in an integral calculus class. The professor quantifies the length
of the table of integrals needed, number of safe transformations needed to be
built into the program, and the number of heuristics transformation needed.
The program is surprisingly simple. An instructor can let the students
know that there are only around 50 things to know in order to completely master
integration.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6034artificialintelligencefall2010/lecturevideos/lecture9constraintsvisualobjectrecognition/
Time: 39:35 to 43:37
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video explains how computer facial
recognition works. The professor writes down an integral that is to be
maximized. Then how demonstrates that it works with a computer program
that he wrote. The math is simple to understand and the application will
excite all students. This is a great way to get students interested in
both calculus and computer science.
Course Topic:
Area Under a Curve
Video Link: https://www.youtube.com/watch?v=fIB5AE4SRN4
Time: 49:22 to 50:29
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video describes how the area under a
curve is used to find the hydraulic radius of a stream. He shows the
typical stream bed and the water above it and how the curve is not just a simple
parabola, but must be determined by arduously collecting data. This could
be used to discus why using area approximations is necessary. The
application is very real life in that it looks at actual surveys that the
professor has done in the field.
Video Link: http://oyc.yale.edu/geologyandgeophysics/gg140/lecture6
Time: 3:45  6:56
University: Yale
Course: The Atmosphere, the Ocean, and Environmental
Change
Professor Name: Ronald Smith
Teaching Ideas: This video shows the graphs of the emitted
radiation vs. the wavelength. He sketches what the graph looks like for a
cool temperature, and intermediate temperature, and a height temperature.
He describes the total radiation as the integral under the curve and the peak
wavelength which is the wavelength that give the greatest radiation. The
professor does not do any calculus and no equations are shown, but this example
can show how calculus will be used.
Video Link: http://oyc.yale.edu/geologyandgeophysics/gg140/lecture17
Time: 44:20  46:28
University: Yale
Course: The Atmosphere, the Ocean, and Environmental
Change
Professor Name: Ronald Smith
Teaching Ideas: This video shows the graphs of the amount of
solar radiation throughout the year for four different latitudes. The
professor notes that even though the poles have a greater maximum amount of
solar radiation, their total radiation which is the area under the curve is much
less, hence it is colder at the poles. There is no actual calculus done in
this clip, but the students can easily imagine how the integral would be used to
find the total annual radiation.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture10
Time: 38:03  39:35
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays graphs of the
number of people in more developed and less developed regions vs. the age
groups. The professor notes that the area under the curve is the total
population which is clearly larger for less developed regions. It is also
interesting to note that the age distribution is skewed right for the less
developed regions and relatively uniform for the more developed regions.
The graphs are shown vertically which gives an instructor an excuse to remind
students the difference between integrating with respect to x (dx) and y (dy).
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture12
Time: 24:00  24:40
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays the graph of the world
population growth over time. Although the professor does not state it, the
area under the curve represents the total population growth. The graph is
on a grid, so it naturally leads itself to using rectangular approximations to
the integral.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture20
Time: 68:49 to 71:33
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the constant in front of
the exponential associated with the Normal density function. The professor
starts with one integral identity and then compares it to the Normal function to
easily find the constant. Not actual integration is done, but the geometry
and the manipulation of the integrand can be a helpful reminder of the basics of
intagration.
Video Link:
https://www.youtube.com/watch?v=rNfR9G3hPa0&list=PL0A0E275BC354C934&index=13
Time: 11:45 to 12:56
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers' Unnamed Guest Speaker
Teaching Ideas: This video goes over the process of
computing the discharge for a stream. The professor's guest speaker goes
through the approximation by rectangles to calculate the total. The
speaker does not say the word integral, but it is clear that the integral is
being calculated. The accompanying PowerPoint is very well done and shows
the method of approximating by rectangles well.
Course Topic:
Indefinite Integrals
Video Link:
http://oyc.yale.edu/physics/phys200/lecture1
Time: 38:30 to 43:00
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at starting with a constant
acceleration and finding the position function. The professor goes through
the step by step logic to derive the solution. He does not use the
integral sign, so it would be a good exercise for the students to fill in the
details and write it as one would in calculus using integration twice. The
professor's discussion is very easy to understand and would work well with the
first day of learning about indefinite integration.
Course Topic:
Finding the Constants of Integration
Video Link:
http://oyc.yale.edu/physics/phys200/lecture1
Time: 44:35 to 47:16
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video begins with the solution of
taking two integrals of a constant and then finding all three constants when the
equation represents the position function with constant acceleration to arrive
at the standard x(t) = g/2 t^{2} + v_{0}t + s_{0}.
This is a simple example on where the +C comes up in physics.
Course Topic:
Substitution
Video Link:
http://oyc.yale.edu/physics/phys201/lecture3
Time: 39:02 to 42:37
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses substitution to integrate a
function that represents the electric field produced by a charged infinite
plate. The answer surprisingly does not depend on the distance from the
plate. The professor explains why and also explains why the integral is
necessary to come up with such a conclusion. Every step in the
substitution is explained in the clip. This is an excellent example of
substitution.
Course Topic:
Average Value of a Function
Video Link:
http://oyc.yale.edu/physics/phys201/lecture13
Time: 49:41 to 51:30
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the average power of an LRC
circuit using the average value integral formula. The professor makes use
of the half angle formula to integrate cos^{2}t and the symmetry of the
sin function to show that the integral over two periods is zero. This is a
solid application that demonstrates fundamental principles and discusses
interesting circuit analysis.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture15
Time: 22:40 to 23:42
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the average combined
electric and magnetic force. It quickly uses the average value formula
without showing any of the work to perform the integral. The instructor
can ask the students to fill in the details to arrive at the professor's answer.
Course Topic:
Integration and Logs
Video Link:
http://oyc.yale.edu/physics/phys200/lecture8
Time: 60:42 to 62:24
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the new velocity of a
rocket after having expelled exhaust. It begins after all the algebra has
been done and shows just the calculus that is performed. The professor
integrates dM/M. He skips several steps which gives the students the
opportunity to be asked to fill in all of the steps, an exercise that will be a
challenge but not unreasonable.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture7
Time: 53:49 to 57:57
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video solves the most basic RC
circuit. Although students at this point in their education have not seen
circuits, the instructor explains it well providing them with an excellent
introduction to circuit analysis. The integration is simple and gives
standard exponential growth for the solution. The professor does apply
separation of variables, but students should be able to follow along even if
they haven't seen that technique before.
Course Topic:
Area Between Two Curves
Video Link: http://oyc.yale.edu/ecologyandevolutionarybiology/eeb122/lecture26
Time: 32:18  33:23
University: Yale
Course: The Nature of Evolution: Selection,
Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video shows the graphs of both the
birth and death rates vs. population density on the same set of axes and on a
new set of axes shows the difference between the birth and death rates.
The professor explains that the intersection of the two curve is the carrying
capacity density. Although the professor does not show this, the area
between the two curves can be interpreted the in total historical population if
the rate of density growth is equal to the time. This could excite the
students during the time in the calculus class when they are just looking at
analytic geometry.
Video Link: http://oyc.yale.edu/geologyandgeophysics/gg140/lecture12
Time: 25:48  30:19
University: Yale
Course: The Atmosphere, the Ocean, and Environmental
Change
Professor Name: Ronald Smith
Teaching Ideas: This video explains why there is a heat
exchange between the equator and the poles. The professor shows a diagram
that shows a graphs of radiation from the sun and emitted radiation by the earth
vs. latitude. The area enclosed by the two curves is the heat surplus and
the area between on the tails is the heat deficit. There are no equations
provided, but the picture is clear and provides strong motivation for needing to
find the area between two curves.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture10
Time: 25:10  27:13
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays graphs of the birth
numbers and death numbers over time for the country of Egypt. It is clear
that there are many more births than deaths. The professor does not do any
calculujs, but students can be asked to interpret the area under the curve.
This is an easy to relate application of area.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture4
Time: 2:58  5:41
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays a very rough diagram of
human population over time showing three distinct stages in human history.
The stages transition due first to farming and agriculture and second to the
industrial revolution. Each segment follows a logistic growth curve.
This is a nice example that shows that carrying capacity can change so the model
must account for that as a piecewise function.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture10
Time: 45:17  47:12 (or 49:10 to see the UN
projections)
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays graphs of the
number of people in more developed and less developed regions vs. the age
groups. The professor notes that the area under the curve is the total
population. Moreover, he notes that after another generation even if the
fertility rate magically becomes two, the shape will get from being
approximately a triangle to approximately a rectangle and the population will
still double. In actuality the rate is over 2 and there will be an
additional contribution to the future population. An instructor can sketch
some graphs that represent these various cases and analyze the future population
using integration.
Course Topic:
Derivative of the Inverse Function
Video Link: http://oyc.yale.edu/geologyandgeophysics/gg140/lecture30
Time: 13:42  15:36
University: Yale
Course: The Atmosphere, the Ocean, and Environmental
Change
Professor Name: Ronald Smith
Teaching Ideas: This video starts with the equation that
relates the earth's temperature to the flux (radiation from the sum per unit
area of the earth). The professor takes the derivative using differentials
and then flips it around to get the derivative of the inverse function.
The derivation is easy to follow and provides an alternative approach to finding
the derivative of the inverse. The application is within the section on
global warming, but that is not evident from the video. The instructor
will need to give the student the premise of the video clip and let them know
that this does not include feedback and CO2.
Course Topic:
Derivative and Integral of an Exponential Function
Video Link:
http://oyc.yale.edu/physics/phys201/lecture8
Time: 26:42 to 29:13
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video performs most of the derivation
of the work done by a capacitor and resister in a circuit. The professor
carries out the derivation taking the derivative of the exponential curve that
corresponds to the charge. Then he leaves it as an exercise to find the
work that is the integral of the square of the current. An instructor can
ask the students to finish up the derivation which will involve a basic
usubstitution to integrate an exponential. This has the potential to work
well with the topics of calculus of exponential functions.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture12
Time: 22:05 to 23:59
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the energy in an LR
circuit which turns out to be an exponential. The professor uses
usubstitution without showing the work. Then he explains that the stored
energy in an inductor drives a current. This is a nice example of
integration of exponentials using usubstitution.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture21
Time: 63:22 to 67:19
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the normalized state
function corresponding to a general exponential function. The integral
involves simple substitution and the students should all understand what the
professor is doing on the board.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture23
Time: 60:59 to 65:52
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the constants that arise
from solving Schrodinger's equation for particle going from a lower potential to
a higher one. The professor sets the derivatives of the two parts equal to
each other and solves. He uses the chain rule for exponential functions.
The calculations are simple, but involve the imaginary number i. At the
end of the clip the professor explains that this indicates that if the energy is
high enough to get over the barrier, sometimes it won't make it.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture17/
Time: 30:30  32:25
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video works out the math to compare the
current and the voltage in a capacitor driven circuit. The math involves
taking the derivative of an exponential function. It is simple chain rule
and the professor calmly and clearly goes through the steps. The only
challenge will be for the students to be able to relate to variables that are
not x and y.
Video Link:
Video Link: http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture22/
Time: 27:50  31:34
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video works out the total energy
provided by the source in an RC circuit. The math involves integrating an
exponential function using usubstitution which the professor does in his head.
This is a clearly presented use of exponentials and integration.
Course Topic:
Hyperbolic Functions
Video Link:
http://ocw.mit.edu/courses/physics/8286theearlyuniversefall2013/videolectures/lecture12noneuclideanspacesopenuniversesandthespacetimemetric/
Time: 50:15  52:51
University: MIT
Course: The Early Universe
Professor Name: Alan Guth
Teaching Ideas: This video integrates an integrand of the
form 1/root(1+kx^{2}) an obtains a hyperbolic sin (sinh x) function that
gives the size of the universe as a function of time, showing the universe grows
without bound. This disproves the Big Crunch theory and demonstrates the
accelerating cold dark theory. This can also be used when doing inverse
trig substitution to show that an alternative method of integration is with an
arctan substitution.
Video Link:
https://www.youtube.com/watch?v=dRR8mu5ISLk&index=14&list=PLE73AA240E8655D16
Time: 4:02  7:43
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video goes over the quantum physics of
a double square well separated by a finite potential with infinite outside
boundaries. The professor writes down the differential equations and
states but does not derive the fact that the solution of the inside portion is a
hyperbolic function. This is a difficult but deep application of
hyperbolic functions. It is the math behind the derivation tunneling.
Course Topic:
Separable Differential Equations and Exponential
Growth
Video Link: http://oyc.yale.edu/chemistry/chem125b/lecture1
Time: 38:20  38:50
University: Yale
Course: Freshman Organic Chemistry II
Professor Name: Michael McBride
Teaching Ideas: This video shows the differential equations
that are used to model zero, first and second order reactions. The
differential equations are all separable and simple to solve. Students can
be asked to solve each of them. The challenge for them will be to accept
that it is ok to have [A], the concentration of A, be a variable.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture12
Time: 15:30  16:13
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays the graph of the world
population over time. The professor notes that the rate of growth has
increased over time so rather than it being an exponential growth, it is a hyper
exponential growth. In a calculus class or differential equations class,
this can be modeled by an equation such as dx/dt = kxt which can be solved by
separation of variables. The professor does not go into this detail, but
it would be a good exercise for students to play with different possible models.
Video Link: http://oyc.yale.edu/ecologyandevolutionarybiology/eeb122/lecture26
Time: 3:01  5:19 (or to 5:55 if there is time to show the
doubling time derivation)
University: Yale
Course: The Nature of Evolution: Selection,
Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video shows the mathematics behind
exponential growth in the framework of population growth in a biology class.
The professor goes through each calculus step just as a math professor would.
This is a great reinforcement of what is done in the calculus class. After
the derivation, if time permits the professor continues to derive the doubling
time formula of 0.69/r.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture31
Time: 32:37 to 35:59
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video derives the integrated rate law
for a first order reaction. The derivation involves integrating 1/[A]
d[A]. The example is the simplest example of a ln integral and uses
chemical notation. Every student should be able to understand what it
represents al long as they have a minimal level of chemistry knowledge.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture32/
Time: 29:38 to 32:19
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video derives the integrated rate law
for a second order reaction. The derivation involves integrating 1/[A]^{2}
d[A]. The professor goes through all the steps and finally explains that
the result can be see as a line with y replaced by 1/[A].
Course Topic:
Volume of Revolution
Video Link:
http://oyc.yale.edu/chemistry/chem125a/lecture12
Time: 7:00 to 8:30
University: Yale
Course: Freshman Organic Chemistry
Professor Name: Michael McBride
Teaching Ideas: This video displays the various orbitals of
electrons. They are almost all solids of revolution. Although the
video shows no mathematical formulas, we can stress to the students that finding
these volumes is instrumental in answering questions that arise in chemistry
such as density and what molecular configurations are possible.
Course Topic:
Work
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 12:27 to 13:31
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video gives a clear explanation
that the work done by a force a distance d is Fd and that this is the distance
in kinetic energy from beginning to end. This will give a reason behind
the definition that is just presented as something to memorize in calculus.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 19:25 to 22:44
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the work done by
stretching a spring. The professor begins with F = ks and then does the
full derivation using rectangles just as a calculus instructor would do in a
calculus class. Instead of integrating ks, the professor treats F as a
generic function of position and presents the integral definition of work.
This will reinforce what is done in that section of calculus.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 40:28 to 43:00
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the conservation
of energy to show that the work expression that includes both gravity and
pulling a spring in combination is a constant. An instructor may need to
fill in the detail of the integration that the professor left out. This is
a direct extension of the standard work calculations done in calculus.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture9
Time: 52:55 to 55:50
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the work done to
take a pendulum that begins vertical and bring it to an angle θ_{0}.
The professor goes through all the steps of the process. This is a clear
example of the use of calculus in physics.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture22
Time: 56:32 to 59:20
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
work done by a gas along an isotherm. The derivation is easy to follow and
is in line with what the calculus textbooks show. This can replace the
corresponding piece of a calculus lecture on work done by gas.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture23
Time: 39:00 to 42:21
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
work done in an adiabatic process. The professor integrates something with
a power in the denominator using the power rule. This is a good time to
stop and remind students that just because there is a denominator, it does not
mean that the integral is a ln.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture2
Time: 67:42 to 69:41
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the work done by an
electric field on a dipole due to torque. The integration is very simple,
but the application is that an electric field can cause a rotation such as
spinning a tire in an electric car. The professor does not explicitly
describe the application, but the instructor can ask the students to come up
with some applications.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture7
Time: 32:31 to 33:38
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the work done to charge up
a capacitor. The integration is very simple as is the full derivation.
This would be a quick application to show students when talking about work.
If you play the video to 35:30 more derivation occurs and relates the energy to
the volume of the space between the capacitor plates.
Course Topic:
Center of Mass
Video Link:
http://oyc.yale.edu/physics/phys200/lecture8
Time: 16:27 to 18:46
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at a one dimensional
object of constant density and derives the obvious formula that the center of
mass is at the center of the line segment. The professor goes through all
the steps of breaking it apart and adding up all of the masses times the
distances. Although this is a very simple example, it serves to explain
the essence of the integral.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture8
Time: 23:42 to 28:33
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the center of mass of
a triangle with constant density. The professor uses symmetry for the
ycoordinate and uses integration for the xcoordinate. This is a very
standard example that one would see in a calculus class. The professor
give the full derivation and can show the students that the exact same math will
be in their physics class. The professor uses similar triangles which
serves as a nice reminder to the students that this technique occurs outside of
math class.
Course Topic:
Moment of Inertia
Video Link:
http://oyc.yale.edu/physics/phys200/lecture9
Time: 63:22 to 66:02
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the moment of
inertia of a disk about its center. The professor goes through each
classic calculus step to solve this problem. His explanation is easy to
understand so this clip can serve as an effective example of this concept.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture9
Time: 67:31 to 69:18
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the moment of
inertia of rod about its endpoint. The calculus is easy and the
explanation is clear. This can be an effective motivator for exploring
this topic.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture9
Time: 70:07 to 71:53
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video is a continuation of the
above video, but calculates the moment of inertia of rod about its center.
The professor uses symmetry to assist in evaluating the integral explaining that
the integral of an even function from a to a is twice the integral from 0 to a.
This is a classic use of symmetry in integration.
Course Topic:
Integrating Powers of Trigonometric
Functions
Video Link:
http://oyc.yale.edu/physics/phys201/lecture21
Time: 46:24 to 47:19
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video calculates the normalized
probability function that corresponds a wave equation. The mathematics
involves taking the integral of the cos2 function over L periods. The
professor does not do the work to solve this integral, so it would be an good
exercise to have the students work it out to verify the solution.
Course Topic:
Inverse Trigonometric Substitution
Video Link:
http://ocw.mit.edu/courses/physics/8286theearlyuniversefall2013/videolectures/lecture9thedynamicsofhomogeneousexpansionpartv/
Time: 22:45  25:13
University: MIT
Course: The Early Universe
Professor Name: Alan Guth
Teaching Ideas: This video goes through the steps of inverse
trigonometric substitution in order to find equations involved in the Big Crunch
theory of the universe. The professor integrates cos(θ) as cos(θ)
by mistake, which makes for a learning opportunity. The physics is not
shown, so we will have to explain that this is for the Big Crunch theory and
that the answer y = 1  cos(θ) shows that at the beginning the universe
had no volume and after one period, we will be back to no volume (The Big
Crunch). At 33:15 the professor shows the curve that is formed graphically
and goes over the physical interpretation. It turns out that the most
recent evidence shows that the Big Crunch theory is incorrect and in fact we
will have an accelerating universe finishing off with a cold dark lonely
universe instead.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture3
Time: 22:57 to 28:20
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the integral of 1/(x^{2}
+ a^{2})^{3/2} using inverse trigonometric substitution.
At the beginning, the professor asks his Yale students how to solve this
integral and none of the students can do it. This is a good chance to let
the students know that inverse trigonometric substitution should not be taken
lightly. At the end the professor explains that this gives the electric
field strength a distance a from a charged wire. This is an important
application in physics and the professor goes through every step of the
integration.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture9
Time: 42:33 to 45:57
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
magnet field at a point above an infinite wire. The integrand turns out to
be the same as the video above. The professor does not solve it, since he
did it six lectures ago. One possibility would be to ask the students to solve
the integral. It is an improper integral but that should not be a big issue.
Course Topic:
Logistics Growth
Video Link:
https://www.youtube.com/watch?v=mRtw4UOwyCM&index=5&list=PL2CD836B66D3CEBED
Time: 2:22  7:00
University: UC Berkeley
Course: Biology 1B (2nd Semester Biology)
Professor Name: Alan Shabel
Teaching Ideas: This video compares the exponential growth
model and the logistics growth model. It exhibits the differential
equations but does not solve them. It only shows the "SCurve" graph.
The last minute of two of the time range should be enough to get the main point
of logistics growth. This can be used either in a calculus course or a
differential equations course, particularly on autonomous differential
equations.
Video Link: http://oyc.yale.edu/molecularcellularanddevelopmentalbiology/mcdb150/lecture4
Time: 2:58  5:41
University: Yale
Course: Global Problems of Population Growth
Professor Name: Robert Wyman
Teaching Ideas: This video displays a very rough diagram of
human population over time showing three distinct stages in human history.
The stages transition due first to farming and agriculture and second to the
industrial revolution. Each segment follows a logistic growth curve.
This is a nice example that shows that carrying capacity can change so the model
must account for that as a piecewise function.
Video Link:
https://www.youtube.com/watch?v=2H6mltgONi8&index=36&list=PL8A25592E6D32C753
Time: 53:45 to 57:39
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video introduces the sigmoid function
which is used in neural networks to find the optimal weights of the network.
The sigmoid function is just the logistics growth curve. The professor
states that it follows y' = y(1y) but does not derive it. The instructor
can pause it there and have the students verify this fact.
Course Topic:
L'Hopital's Rule
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6034artificialintelligencefall2010/lecturevideos/lecture11learningidentificationtreesdisorder/
Time: 27:50 to 30:36
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video defines the disorder of a set in
terms using log_{2}. The function is written down and looks at a
special case which when plugged looks like xlog_{2}x. The
professor names L'Hopital's rule and describes how it works, but does not go
through the details. The instructor can pause it and ask the students to
fill in the details. The instructor will also need to explain that the
professor is using this in order to decide what question to ask first in order
to get at an answer as quickly as possible when there are many possible
questions to ask. As a side note, this formula is also used to measure
diversity in biology. This is a relevant application of logarithms.
Course Topic:
Improper Integrals
Video Link:
http://ocw.mit.edu/courses/physics/8286theearlyuniversefall2013/videolectures/lecture23inflation/
Time: 42:00  45:50
University: MIT
Course: The Early Universe
Professor Name: Alan Guth
Teaching Ideas: This video uses an improper integral to
calculate the event horizon of the universe. This is the distance such
that an object must be from us so that light from it will never reach us due to
the acceleration of the universe and the speed of light. This is a
fascinating application of improper integrals that will encourage student
interest.
Video Link:
http://oyc.yale.edu/economics/econ25211/lecture8
Time: 40:24 to 40:43
University: Yale
Course: Financial Markets
Professor Name: Robert J Shiller (Nobel Laureate)
Teaching Ideas: This video presents the equation for the
Present Discounted Value when payments are coming in continuously.
If time permits, the minute of lecture before this clip shows the infinite
series for annual and biannual payments which is an infinite series. This
could be used to introduce how an infinite series becomes an improper integral.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture20
Time: 61:12 to 62:41
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video writes down the rule for a
function to be a probability density function. The professor shows that
this is the integral from negative infinity to infinity of the density function
must be 1. There are no examples given, but the rule is a nice application
of improper integrals.
Video Link:
http://ocw.mit.edu/courses/chemistry/5112principlesofchemicalsciencefall2005/videolectures/lecture3waveparticledualityofradiationandmatter/
Time: 4:59 to 8:08
University: MIT
Course: Principles of Chemical Science
Professor Name: Sylvia Ceyer
Teaching Ideas: This video calculates and graphs the total
energy of an electron nucleus classical system with the electron in orbit around
the nucleus. The calculation involves a simple improper integral that the
professor does in her head. After the clip the professor explains that
when electromagnetism is included the model becomes impossible. Students
can be asked to fill in the details of the integration and talk about the
infinity in the bound.
Course Topic:
Tests for Convergence of a Series
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6006introductiontoalgorithmsfall2011/lecturevideos/lecture4heapsandheapsort/
Time: 40:59 to 45:23
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video shows the computer work required
to go through an algorithm. The derivation involves a piece in the form:
Sum of (k+1) / 2^{k}. The professor does not show this but the
punch line of the proof is that the series converges so is bounded by a
constant. Students can be asked to prove the convergence of the series
which can easily be done with a limit comparison test or a ratio test or even an
integral test which will involve integration by parts. The instructor will
have to explain the context since the clip does not go over the details of the
algorithm.
Course Topic:
Taylor Polynomials and Series
Video Link:
http://oyc.yale.edu/economics/econ251/lecture7
Time: 25:50 to 29:09
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video proves the rule of 72 which says
that the time to double your money is about 72/(100i) where i is the interest
rate as long as i is not that far from 7%. In the derivation, the
professor uses the Taylor expansion from ln(1+i) finding the first three terms.
This is a pretty simple example of Taylor polynomials being used and can be show
to the students why the Taylor polynomial his helpful for basic estimations.
Video Link:
http://oyc.yale.edu/economics/econ251/lecture22
Time: 6:25 to 8:58
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video goes over Bernoulli's St.
Petersburg Paradox which looks at an infinite expected value where people are
not willing to pay much for that bet. Instead there is a utility function
that is logarithmic. He goes over this infinite series and realizes it as
a logarithm. He does not give the details of the power series that gives
this log, but it is something that can be given to the students as an exercise.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture5
Time: 66:33 to 68:02
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video gives the full statement of
the Maclaurin Series formula just as a calculus instructor would give. No
applications are provided, but the explanation is easy to follow. If you
watch until 69:38 you will see the example e^{x}. This provides
reinforcement to what is done in a calculus class.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 0:59 to 8:05
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video gives an argument for the
McLaren Series. The professor starts with the constant approximation. then
moves on to the linear approximation and continues with the quadratic
approximation. Finally he writes down the full series. The professor
state that he is doing it in the way physicists do it, but it is no different
from the way it is done in calculus class. This can replace the
instructor's introduction of McLaren Series.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 8:58 to 12:00
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the McLaren series
for the function 1/(1x). The professor does not refer to any physics, but
the mathematics he does is the same as what students see in a calculus class.
Next the professor uses it to show what is happening with 1/(1  0.1).
This is a nice example of showing numerically what is happening with the
geometric series. If time permits, the continuation through 14:20
discusses the convergence of this series.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 17:30 to 18:37
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the McLaren series
for e^{x}. He does this quickly and clearly. Although this
is a physics class, he just does the math here and does not go into how it can
be used in physics.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 20:15 to 22:09
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the McLaren series
for cos x. Like the clip above, the professor does this quickly and
clearly. Also as above, he just does the math here and does not go into
how it can be used in physics.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture16
Time: 25:21 to 27:13
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses McLaren series to
derive Euler's formula. Like the clips above, the professor does
this quickly and clearly. Also as above, he just does the math here and
does not go into how it can be used in physics.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6002circuitsandelectronicsspring2007/videolectures/lecture7/
Time: 32:55 37:21
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses the small signal method and
the mathematical method of Taylor series to find the response to a signal.
The professor explains that the small signal makes it acceptable to ignore the
nonlinear terms. The math is done theoretically, but the application is
very real world.
Video Link:
https://www.youtube.com/watch?v=ZJlh1TUjDVI&list=PLE73AA240E8655D16&index=8
Time: 21:41  25:12
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video explains that the difference
between a harmonic oscillator and an anharmonic oscillator is that the harmonic
oscillator is just the first degree Taylor polynomial instead of the full curve.
The professor draws the graph and explains the physical difference between the
two. This is a nice geometric display of the first degree Taylor
approximation.
Course Topic:
Series Expansion of a Binomial
Video Link:
http://oyc.yale.edu/astronomy/astr160/lecture10
Time: 4:30  7:19 and 8:30  10:22
University: Yale
Course: Frontiers and Controversies in
Astrophysics
Professor Name: Charles Bailyn
Teaching Ideas: This video shows is an application using the
Taylor polynomial to realize where Newtonian and Post Newtonian physics differ.
The professor does not actually show the calculus, but it would not be difficult
for the students to fill in the details. This could also be used in a first
quarter calculus course as an application of the tangent line approximation to a
curve. Then at 8:30, the derived formula is applied to gamma the
relativistic factor.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture7
Time: 64:24 to 66:43
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses the second degree Taylor
expansion of the binomial to derive the classic formula from physics that the
potential energy of a object subject to gravity is GMm/R^{2}. Many
students know the inverse square relationship, but this clip shows them why and
convinces them that Taylor Series can be useful.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture14
Time: 64:40 to 67:58
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video works out the energy of a
particle by looking at the power series expansion of its energy formula.
The second term of the Taylor series is 1/2 mv^{2}, the kinetic energy.
The first term is mc^{2} which is the rest energy and where Einstein
came up with his famous equation. This is a pretty easy derivation and may
be the first time that students understand the famous equation. Students
will have a great sense of why power series are important.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture17
Time: 15:13 to 16:46
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses the binomial series to
demonstrate how a spherical mirror can approximate a parabolic mirror. The
math is simple to follow and the physics is relevant for those who are
interested in optics.
Course Topic:
Quadric Surfaces
Video Link: http://oyc.yale.edu/geologyandgeophysics/gg140/lecture6
Time: 17:00  18:53
University: Yale
Course: The Atmosphere, the Ocean, and Environmental
Change
Professor Name: Ronald Smith
Teaching Ideas: This video shows the dynamics of a plume of
air that is released and shot out horizontally by a wind if there is a steady
state output of pollutants. After a fixed number of minutes, the plume
will be in the shape of a paraboloid with vertical cross sections as circles and
horizontal cross sections as parabolas. No equations are given, but the
hand drawn pictures are clear and the professor nicely demonstrates that drawing
cross sections is a good graphing technique.
Video Link:
https://www.youtube.com/watch?v=2H6mltgONi8&index=36&list=PL8A25592E6D32C753
Time: 33:12 to 35:16
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video finds the best weights of a
neural network and explains that the error function is a paraboloid. The
professor shows that the gradient descent method is what is used to locate the
unique global minimum of the paraboliod. This can either be shown when
introducing quadric surfaces or gradients.
Course Topic:
Contour Diagrams
Video Link:
http://oyc.yale.edu/chemistry/chem125a/lecture3
Time: 19:00  16:26
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video shows an application of a contour
diagram to understanding energy states and activation energies. The
professor clearly demonstrates how the gradient curve (although he does not use
these words) demonstrates what must occur in order for a reaction to take place.
This can be used either when introducing functions of several variables or when
discussing gradients or extrema of multivariate functions.
Video Link:
https://www.youtube.com/watch?v=z_fOdaNTppI&list=PL0A0E275BC354C934&index=8
Time: 46:28 to 48:50
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video shows how to look at a topo map
to analyze where debris flows have occurred in the past. The professor
explains how these occur during hurricanes. The maps are clearly labeled
and the debris flows are striking. He notes that 400 people were killed by
just one event in New Orleans. This can be shown when introducing the
graph of a function of two variables.
Course Topic:
Spherical Coordinates
Video Link: http://oyc.yale.edu/chemistry/chem125a/lecture9
Time: 24:40  27:57
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video uses spherical coordinates to
simplify the Schrodinger equation for the one electron hydrogen atom. The
professor explains how the use of spherical coordinates allows us to write the
Schrodinger equation as a product of three functions, each a function of a
single variable. He uses r instead of r
and switches the roles of theta and phi, so the students will have to be told
that in applications, the variable names are not standardized.
Video Link:
http://ocw.mit.edu/courses/chemistry/5111principlesofchemicalsciencefall2008/videolectures/lecture4/
Time: 41:54  44:50
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video argues that using spherical
coordinates is the better approach to looking at the hydrogen atom compared to
rectangular coordinates. The professor shows the spherical plane, but uses
the variable naming convention that switches the angles. Then she shows
the Schrodinger equation in spherical coordinates. She doesn't do anything
with them other then say that it happens in more advanced course. This is
a clear example of spherical coordinates being used for high level topics.
Video Link:
http://ocw.mit.edu/courses/chemistry/5112principlesofchemicalsciencefall2005/videolectures/lecture30crystalfieldtheorycont./
Time: 19:20 to 21:30
University: MIT
Course: Principles of Chemical Science
Professor Name: Christopher Cummins
Teaching Ideas: This video uses spherical coordinates to
express the orbital shells of electrons. The professor defines the
spherical coordinate system with the angles switched compared to how it is done
in calculus and he uses r instead of ρ. He explains that the
dorbitals can be written independent of the radius. This is a nice
application of spherical coordinates that will be familiar to students who have
had chemistry.
Course Topic:
Partial Derivatives
Video Link: http://oyc.yale.edu/ecologyandevolutionarybiology/eeb122/lecture11
Time: 38:07  39:00
University: Yale
Course: The Nature of Evolution: Selection,
Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video looks at the partial derivative
partial derivative of fitness (number of offspring) of a male with respect to
further survival. It just shows the graph based on age but not the
equations. The graph is given and shows that "after the age of 46
evolution doesn't care if you are there anymore." This gives a very
meaningful lesson based on partial derivatives.
Video Link:
http://ocw.mit.edu/courses/physics/8333statisticalmechanicsistatisticalmechanicsofparticlesfall2013/videolectures/lecture4thermodynamicspart4/
Time: 9:00  10:59
University: MIT
Course: Statistical Mechanics
Professor Name: Mehran Kardar
Teaching Ideas: This video uses the fact that the mixed
partial derivatives are independent of order to prove one of Maxwell's results
about thermodynamics relating energy, temperature, force, position, momentum and
enthalpy. The physics will be way over the heads of the students, but it
is helpful for them to see what graduate level physics looks like.
Video Link:
http://oyc.yale.edu/economics/econ251/lecture2
Time: 48:50 to 52:38
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video presents the diminishing marginal
utility which looks at the partial derivatives with respect to x and y and
notices they are negative so diminishing. This is a very simple example of
an application of partial derivatives.
Video Link:
http://oyc.yale.edu/economics/econ251/lecture3
Time: 31:25 to 32:02
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video looks at the utility functions
and explains that the marginal utility of x (partial derivative with respect to
x) divided by the price of x equals the same quotient in y. The lecture is
somewhat scattered, but it is a common example of using partial derivatives.
Students will need to be told what the professor is talking about.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture6
Time: 2:30 to 4:28
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video defines the partial derivative
with respect to x. Although this is a physics class, this clip could have
just as well been taken from a calculus class. This could be shown as a
reinforcement to the motivation behind partial derivatives or it can just
replace the in class lecture on the subject.
Video Link:
http://oyc.yale.edu/physics/phys200/lecture6
Time: 5:15 to 8:58
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows an example of taking both
the first and second partial derivatives of a polynomial function in two
variables. The professor clearly shows the notation and the solutions and
makes a point at the end to explain, but not prove, that the mixed partial
derivatives are equal. This could come right out of a math class, but
since it is a physics class students will realize how important this concept is.
The proof is shown over the next 7 minutes until 16:30.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture14
Time: 10:31 to 12:35
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the wave equation which
is a partial differential equation. The clip starts in the middle of the
derivation, but the beginning is slow and may not be worth the class time to
show it all. The instructor will want to give a brief introduction to get
the students caught up to where the professor begins.
Video Link:
https://www.youtube.com/watch?v=V5T_ywvofpE&list=PLE73AA240E8655D16&index=22
Time: 17:42  21:32
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives a quantum mechanical
formula for square of the momentum operator in the radial direction. This
involves taking derivatives with respect to r. The math is the appropriate
level, but the pace is lightning fast. The physics will be far too
difficult for the students to relate to, but some enjoy seeing high level work.
Course Topic:
The Gradient Vector
Video Link:
http://oyc.yale.edu/physics/phys201/lecture5
Time: 61:26 to 63:14
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shows that the gradient of the
electric potential gives back the electric field. The professor computes
the partial derivatives and packages them as a vector field after taking the
gradient. The professor never states the word "gradient" but the
instructor can tell the student that that this is what happened.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture6
Time: 40:24 to 42:42
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains that the electric field
points in the greatest change in potential. Then the professor mentions
that going in the direction in the gradient vector will be most efficient.
Finally, he shows the physics formula for the directional derivative. The
notation looks different and the words "directional derivative" are not used,
but it would be a good question of the students to see if they can answer the
question of what he has just defined.
Video Link:
http://ocw.mit.edu/courses/physics/8224exploringblackholesgeneralrelativityastrophysicsspring2003/lecturenotes/5einsteinsfieldequations/
Time: 9:48 to 10:46
University: MIT
Course: Exploring Black Holes: General
Relativity and Astrophysics
Professor Name: Edmund Bertschinger
Teaching Ideas: This video presents the gravitational
potential function in terms of the gradient vector. The professor doesn't
do much in this clip, but it does demonstrate a use of the gradient vector that
is not standard in calculus textbooks.
Video Link:
https://www.youtube.com/watch?v=2H6mltgONi8&index=36&list=PL8A25592E6D32C753
Time: 35:16 to 41:03
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video finds the best weights of a
neural network using the method of gradient descent. The professor goes
through the calculus derivation and finally summarizes the process. This
is a high level clip that will interest students who are curious about how
computers perform tasks such as face and voice recognition.
Course Topic:
Chain Rule for Partial Derivatives
Video Link:
http://oyc.yale.edu/physics/phys200/lecture6
Time: 19:34 to 24:10
University: Yale
Course: Fundamentals of Physics I
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the
derivative of the kinetic energy with respect to time in the context of finding
the work done. The professor begins with the standard kinetic energy
theorem in two variables and then takes a partial derivative with respect to t
on both sides. In the derivation he makes use of the chain rule with the
composition function R^{1} > R^{2} > R^{1}. The
explanation if clearly stated and can be used to introduce the chain rule or to
give an example of this type of chain. The clip can be ended at several
points along the way if time is a big issue.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture14
Time: 14:18 to 17:38
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video proves that the wave equation has
solution of the form F(x  vt). The professor uses the chain rule for
partial derivatives multiple time since the wave equation is a second order
partial differential equation. The calculus is at the appropriate level
for first year students learning about functions of two variables.
Video Link:
http://oyc.yale.edu/physics/phys201/lecture14
Time: 59:15 to 62:04
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the relationship between
the magnitude of the electric force vector, E, and the magnetic force vector, B.
It uses pieces of the wave equation which involves partial derivatives.
The derivation is easy to follow and can be shown on the first day of a partial
differential equations class.
Video Link:
https://www.youtube.com/watch?v=KZe6LqSMW9Y&list=PLE73AA240E8655D16&index=20
Time: 30:24  33:52
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video derives the quantum mechanical
angular momentum along the zaxis in spherical coordinates. The derivation
involves the transformation equations for spherical coordinates and explicitly
uses the chain rule for multivariable functions. The physics will be too
difficult for the students to relate to, but some enjoy seeing high level work.
The math done is at just the right level for advanced calculus students.
Course Topic:
Finding Extrema for Multivariate Functions
Video Link: http://oyc.yale.edu/chemistry/chem125a/lecture9
Time: 46:30  47:55
University: Yale
Course: Freshman Organic Chemistry I
Professor Name: Michael McBride
Teaching Ideas: This video applies the technique of taking
the partials and setting them equal to 0 to find the extrema for the 2p orbitals
of a hydrogen atom.
Video Link:
http://oyc.yale.edu/economics/econ159/lecture4
Time: 24:10 to 28:18
University: Yale
Course: Game Theory
Professor Name: Ben Polak
Teaching Ideas: This video gives the definition of what is
means to be a "best strategy" in game theory. The definition just is the
definition of the maximum value of a two variable function. The professor
does not do any computation, but the definition can help students see where
extrema of multivariable functions can be useful. The application that
the professor goes over earlier is in soccer: kick left, middle, or right.
Course Topic:
Constrained Optimization
Video Link:
http://oyc.yale.edu/economics/econ251/lecture3
Time: 3:55 to 5:11
University: Yale
Course: Financial Theory
Professor Name: John Geanakoplos
Teaching Ideas: This video writes down the utility function
for two goods subject to two constraints that are based on the amount of product
available. The professor does not solve it here, but it is a great class
exercise to ask them to solve it using Lagrange Multipliers. The solution
is not very difficult, but looks tough due to having six variables. The
professor solves it in the next five minutes, but spend a long time with a not
that clear or simple solution.
Course Topic:
LaGrange Multipliers
Video Link: http://oyc.yale.edu/ecologyandevolutionarybiology/eeb122/lecture32
Time: 5:30  8:51
University: Yale
Course: The Nature of Evolution: Selection,
Inheritance and History
Professor Name: Stephen C. Stearns
Teaching Ideas: This video demonstrates that the optimized
line through the origin for deciding to stop searching for food in a patch of
land and start searching in another one is found by rotating it until it is
tangent to the payoff vs. time curve. This is similar to the search of the
maximum value given a constraint that the method of LaGrange multipliers finds.
This can be used to introduce constrained optimization problems.
Video Link: http://oyc.yale.edu/economics/econ159/lecture14
Time: 28:15 to 33:44
University: Yale
Course: Game Theory
Professor Name: Ben Polak
Teaching Ideas: This video looks at the quantity that a
first firm should produce if that first firm knows how much the competitor will
produce in reaction to the first firm's decision. An equation is presented
and the professor indicates that one could substitute the constraint equation
into the max/min equation and then take a derivative, but another method
is the method of LaGrange multiplier. This is a nice reminder to calculus
students that there are two ways to solve constrained optimization problems.
The professor decides to solve using substitution rather than LaGrange
multipliers. Students can be asked to do the problem, but it is a bit of
alphabet soup. This can also be used as an application of first quarter
calculus.
Video Link:
http://ocw.mit.edu/courses/electricalengineeringandcomputerscience/6034artificialintelligencefall2010/lecturevideos/lecture16learningsupportvectormachines/
Time: 22:10  26:12
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video uses Lagrange Multiplier to help
a computer use artificial intelligence to find the band that separates two
sets. The actual math comes after the clip and is too difficult for
typical first year calculus students, but this humorously presented clip will
help them see an interesting application.
Video Link:
https://www.youtube.com/watch?v=DYpvNFgrBWU&index=5&list=PLkOvqP5rUuNF5G_ooxxmrPJN8U7Kmtm1
Time: 106:30  114:04
University: UC Berkeley
Course: Environmental Economics and Policy
Professor Name: David Zetland
Teaching Ideas: This video uses Lagrange Multiplier to solve
an economics problem. The lecture is very mathematical and proceeds just
as a calculus instructor would, but the professor explains it in the framework
of economic theory instead of pure mathematics.
