Please work out each of the given problems.
Credit will be based on the steps that you show towards the final answer.
Show your work.
PROBLEM 1 Please answer the following true or false.
If false, explain why or provide a counter example.
If true, explain why.
A) If
f(x) = x2 - x
then the tangent line to
f(x) at x = 2 is parallel to the line y
= 3x + 7.
B) If
x2 - 4
f(x) =
x + 2
then f(x) is
continuous at x = -2
C) If a
function f has a limit L at x
= c, and f(c) = L, then the function is
differentiable at x = c.
PROBLEM 2 Explain in
your own words what the difference between continuity and differentiability is.
PROBLEM 3
The graph of y = f(x) is
given below.
A) Evaluate
the following limits if they exist:
i)
ii)
iii)
iv)
B) At what points is f(x)
not continuous?
C) At what points is f(x) not differentiable?
PROBLEM 4
Evaluate the following limits if they exist.
A)
B)
C)
PROBLEM 5 The amount of algae in Lake Tahoe over the past twenty years can be modeled by the function
A(t) = 200 e.06t
where t represents the time in years after 1968 and A
represents the kg of algae in the lake.
A) Is
this model a differentiable function? Explain.
B) Use a
graphing calculator to graph this function.
Over what domain is this a reasonable model?
C) Approximate
(with the assistance of your graph) the slope of the tangent line when t
= 20 and when t = 40.
D) What
do your answers from part C mean in terms of the algae in Lake Tahoe?
PROBLEM 6 Find dy/dx
for the following
PROBLEM 7 Use the limit definition of the derivative to find the derivative of
PROBLEM 8 Your research shows that the profit P in dollars from renting out x bicycles at your bicycle rental shop that you manage is given by
P = -.03x2 + 25x - 40
A. Find the additional profit when rentals increase from 25 to 26 units.
B. Find the marginal profit when x = 25.
Extra Credit:
Write
down one thing that your instructor can do to make the class better and one
thing that you think the instructor should not change.
(Any constructive remark will be worth full credit.)