# First Year Calculus

Course Topic:

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 = mc2.  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.

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.

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.

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:

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.

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:

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.

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.

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 position-velocity-acceleration for an object that is moving with constant acceleration.

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.

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.

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.

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.

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:

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.

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:

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:

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.

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:

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.

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:

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.

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.

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 ω2.  This emphasizes that the derivative of the inside is what makes velocity and acceleration differ from each other and from position.

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:

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.

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.

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.

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.

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.

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.

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.

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.

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.

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".

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 xy-plane 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.

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.

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.

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.

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.

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.

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:

Time:  22:53 to 27:44

University:  MIT

Course:  Introduction to Algorithms

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.

Time:  4:07 to 8:53

University:  MIT

Course:  Introduction to Algorithms

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.

Time:  26:20 to30:16

University:  MIT

Course:  Introduction to Algorithms

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:

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) = x2.  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.

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.

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.

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:

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:

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.

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:

Time:  56:24 to 57:12

University:  Yale

Course:  Fundamentals of Physics I

Professor Name: Ramamurti Shankar

Teaching Ideas: This video finds the anti-derivative 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".

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.

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.

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.

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 z-axis 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.

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 anti-derivative.

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.

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.

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:

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.

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.

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.

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).

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.

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.

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:

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:

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 t2 + v0t + s0.  This is a simple example on where the +C comes up in physics.

Course Topic:

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:

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 cos2t 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.

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:

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.

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:

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.

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.

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.

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.

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:

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:

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 u-substitution to integrate an exponential.  This has the potential to work well with the topics of calculus of exponential functions.

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 u-substitution 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 u-substitution.

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.

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.

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.

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 u-substitution which the professor does in his head.  This is a clearly presented use of exponentials and integration.

Course Topic:

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+kx2) 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.

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.

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.

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.

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.

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.

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:

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:

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.

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.

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.

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 θ0.  The professor goes through all the steps of the process.  This is a clear example of the use of calculus in physics.

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.

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.

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.

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:

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.

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 y-coordinate and uses integration for the x-coordinate.  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:

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.

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.

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:

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:

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.

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/(x2 + a2)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.

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:

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 "S-Curve" 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.

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.

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(1-y) but does not derive it.  The instructor can pause it there and have the students verify this fact.

Course Topic:

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 log2.  The function is written down and looks at a special case which when plugged looks like xlog2x.  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:

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.

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.

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.

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:

Time:  40:59 to 45:23

University:  MIT

Course:  Introduction to Algorithms

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) / 2k.  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:

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.

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.

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 ex.  This provides reinforcement to what is done in a calculus class.

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.

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/(1-x).  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.

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 ex.  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.

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.

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.

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.

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:

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.

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/R2.  Many students know the inverse square relationship, but this clip shows them why and convinces them that Taylor Series can be useful.

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 mv2, the kinetic energy.  The first term is mc2 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.

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:

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.

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:

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.

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:

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.

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.

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 d-orbitals 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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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 R1 -> R2 -> R1.  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.

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.

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.

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 z-axis 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:

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.

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 multi-variable functions can be useful.  The application that the professor goes over earlier is in soccer:  kick left, middle, or right.

Course Topic:

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:

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.

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.

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.

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.