Test#3 Review
“You just keep pushing. You just keep pushing. I made every mistake that could be made. But I just kept pushing” Rene Descartes
Note: These problems are taken from your homework and chapter 5 review test in your textbook.
1.Prove the following identities.
a.b.
2.Show that . This one was done in class.
3.Find the derivative. Simplify where possible.
a.b.
c.
4.State the Extreme Value Theorem, definition of critical values, and the Mean Value Theorem.
5.Find the absolute maximum and absolute minimum values of f on the given interval.
a. [-1,1]. b. on [1,3]
6.Sketch the graph of a function on [-1, 2] that has an absolute maximum but no local maximum.
7.State Rolle’s Theorem and verify that the following function satisfies the three hypotheses. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem.
[0, 9]
8.Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.
on [0,2]
9.Show that the equation has at most two real roots.
10.The graph of the derivative of a function f is shown.
a)On what intervals is f increasing or decreasing?
b)At what values of x doe f have a local maximum or minimum?
c)Sketch the graph of
d)Sketch a possible graph of
11.Given
Find the interval on which f is increasing or decreasing.
Find the local maximum and minimum values of f.
Find the intervals of concavity and the inflection points.
12.Sketch the graph of f that satisfies the following conditions.
13.Sketch the graphand provide the point of inflections (if any).
a.b.
14.Use L’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If I’Hospital’s Rule doesn’t apply, explain why.
a.b.c.
15.A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other into an equilateral triangle. How should the wire be cut so that the total areal enclosed is a) maximum? b) a minimum?
16. Review other word problems we did in class on the concept of optimization.
Solutions
1.This is just using the definitions of hyperbolic functions. See your notes
2.I did this in class, I think. There is an example in the book as well.
3
a)
b)
c)
4.EVT-If f is continuous on [a,b], then f attains an absolute maximum value f(c) and an absolute value f(d) where c and d belong in [a,b]
MVT-Let f be a function satisfying
i) f is on cont. on [a,b]
ii) f is differentiable on (a,b).
Then there exists c in (a,b) so that
which can also be rewritten as
5.a)Absolute max. value: f(1)=ln3 and Absolute min. value: f(-1/2)=ln(3/4)
b)Absolute max. value: and absolute min. value:
6.This is an easy one.
7.Justify why f is continuous, differentiable, and f(0)=f(9). The c value is c=9/4 that satisfy Rolle’s Thm.
8.Justify why f is continuous and differentiable on (0,2). .
9.I think this example was done in class.
10.
a)On what intervals is f increasing or decreasing?
Increasing: (-2,0) and
Decreasing: and (0,4)
b)At what values of x doe f have a local maximum or minimum?
Local max: x=0,
Local min: x=-2, 4
c)Sketch the graph of
Try it
d)Sketch a possible graph of
Try it
11.This is the graph. Find the absolute max by using the derivative.
12.
13.a.
y-int: (0,0)
x-int:
f is odd means it is symmetry along the origin
No asymptote
Critical values: x-0,-1,1
Local Max= f(-1)=2 and Local Min: f(1)=-2
Point of inflection is at (0,0).
b.Do this on your own.
14.a) the limit is 0
b) the limit is ½
c) the limit is
15.The max. area will happen when x=10m.
The min. area occurs when m or approximately 4.35m.