The purpose of this review sheet is to give an overview of what we did this quarter, and to remind you of what you are supposed to know for the final. It’s not enough to just study from this sheet! Go over your notes, the book, the homework and the chapter tests.
These problems are intended to help you study for the final. However, you shouldn’t assume that each problem on this handout corresponds to a problem on the final. Nor should you assume that if a topic doesn’t appear here, it won’t appear on the final.
The final exam is on Thursday (Mar 15) from 8 – 10 in this room. (I will be here at 7:30 if you want to start early). You are allowed one 8 ½ inch by 11 inch piece of paper on which you can put any notes you want to. On the class website, I put a summary of derivatives or you can use the green derivative summary sheet that I passed out.
1. Find the derivative of using the definition of the derivative.
2. Find the derivative of the following functions.
a)
b) c) d) e)
3. Compute the following limits.
a)
b)
4. If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is given by
a) What is the maximum height reached by the ball?
b) What is the velocity of the ball when it is 96 ft above the ground on its way up?
5. If a snowball melts so that is surface area decreases at a rate of 1 cm2/min, find the rate at which the diameter decreases when the diameter is 10 cm.
6. Find the linearization of the function at a = 1.
Has a graph shown below.
a) Find an equation of the tangent line to this curve at the point (1, -2).
b) Let be the point on the curve whose y-coordinate is -2.1. Using linear approximation and part (a), estimate b.
c) Find the coordinates of all points where the tangent line to the curve is horizontal.
8. The function is graphed below. You need not show any work on this problem.
Calculate the following: a)
b)
e) If then
9. The graph of f is shown below.
Use the given graph of f to find the following: a) The intervals on which f is increasing. b) The intervals on which f is decreasing.
c) The intervals on which f is concave downward. d) The intervals on which f is concave upward. e) The coordinates of the points of inflection.
10. For the function
a) Find the intervals of increase or decrease b) Find the local maximum and minimum values
c) Find the intervals of concavity and the inflection points. d) Sketch a graph of f(x) using the above information.
12. Let
a) Find the x-coordinate of all critical numbers of in .
b) Determine if the critical numbers you found are local maxima, local minima, or points of inflection.
c) Locate the global minimum of in the interval
13. A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence.
14. Find f if
Answers
1. Find the derivative of using the definition of the derivative.
2. Find the derivative of the following functions.
a)
b)
c)
d)
e)
3. Compute the following limits.
a) , use L’Hopital’s rule:
Use l’hopital’s rule:
b) , simplify numerator
use L’Hopital’s rule:
4. If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is given by
a) Maximum height occurs when the velocity is zero: b) Find the time to reach 96 ft:
5. cm/s
6. Find the linearization of the function at a = 1.
7. The equation of the curve:
a)
.
b) .
c)
When x = 0, y = 0 – from the graph, the curve does not have a horizontal tangent at this point. For x = -2, . The two points are (-2, 2) and (-2, -2).
8. For the function , calculate the following:
a) -10
b) -1
c) -1
d) 0
e) If then
9. Use the given graph of f to find the following:
c) The intervals on which f is concave downward. d) The intervals on which f is concave upward. e) The coordinates of the points of inflection.
10. For the function
a) From sign chart, we get for ; for
b) Local maximum: (-1, 5); local minimum (1, 1)
c) From sign chart,
for ; for
.
inflection points: (-0.707, 4.24), (0. 3) and (0.707, 1.76) d) .
11. Use information from the first and second derivative tests to sketch a graph of the function
; as a critical number
for x < 1 and for x > 1. Maximum at (1, 0.368)
12. Let a)
These are the critical numbers of
in .
b)
; In the interval , at
From sign chart, for ; for so a local
maxima occurs at , and the point of inflection is at .
13. A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence.
x
y ; Let P represent the total length of fencing required.
ft, x = 1000 ft
14. Find f if
between the two cars changing when car A is 0.3 miles and car B is 0.4 miles from the intersection?
x
y d
Car A: Car B:
Evaluate at the given values.
