Calculus I Exam 4 Review (Chapter 4)
Review problems from pp. 254-256: 1-8, 39, 40, 51-58 plus the following problems. In addition to the problems listed from the chapter review and these problems, all of your homework problems and quiz problems are fair game for the test as well. Please be sure to have your
homework (4.1-4.3, worksheet on 4.5, 4.7) completed and ready to hand in on Thursday.
1.
Find the critical numbers of each function. Give exact values, not decimal approximations . a) 𝑓(𝑥) = 𝑥 3 + 6𝑥 2 − 15𝑥 b) 𝑓(𝑥) = ln 𝑥 𝑥 2 c) 𝑓(𝑥) = 𝑥
2
5
(𝑥 − 5) 3
2.
Verify that each function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. a) 𝑓(𝑥) = 𝑥 3 − 5𝑥 2 + 4 ; [0, 5] b) 𝑓(𝑥) = √36 − 𝑥 2
; [-3, 3]
3.
Verify that each 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.
a) 𝑓(𝑥) = √2𝑥 + 10 ; [-5, 3] b) 𝑓(𝑥) = 𝑥 𝑥+2
; [1, 4]
4.
Use the graph of f shown below to estimate the values of c that satisfy the conclusion of the
Mean Value Theorem on the interval [0, 6]. Justify your answer with a sketch.

5.
Sketch the graph of a function which satisfies the following sets of conditions. a) 𝑓 ′ (𝑥) is positive when 𝑥 < 0 and negative when 𝑥 > 0 ; 𝑓"(𝑥) is always negative. b) 𝑓 ′ (𝑥) is positive when 𝑥 < 0 and negative when 𝑥 > 0 ; 𝑓"(𝑥) is always positive. c) 𝑓 ′ (𝑥) is always positive; 𝑓"(𝑥) is positive when 𝑥 < 0 and negative when 𝑥 > 0 .
Find the absolute maximum and minimum values of f on the given interval.
6.
𝑓(𝑥) = 4𝑥 3 − 12𝑥 2 − 36𝑥 + 9 ; [−2, 4]
7.
𝑓(𝑡) = 𝑡√9 − 𝑡 2 ; [−1, 3]
For each of the following functions, determine a) the intervals of increase or decrease, b) the local maximum and minimum values, c) the intervals of concavity, and d) any inflection points. Use the first and second derivatives and show work.
3
9.
𝑓(𝑥) = √𝑥 (𝑥 + 4) 8.
𝑓(𝑥) = 𝑥 3 + 9𝑥 2 − 600
10.
𝑓(𝑥) = 𝑒 𝑥 + 𝑒 −3𝑥
12. 𝑓(𝑥) = 𝑥 7 ln 𝑥
11.
𝑓(𝑥) = 𝑠𝑖𝑛 2 𝑥 , [0, 2𝜋]

13.
Find the local maximum and minimum values of f using the first and second derivative tests.
Show work.
𝑓(𝑥) = 7 + 3𝑥 2 − 2𝑥 3
14.
Suppose you have 200 feet of fencing and want to enclose a rectangular area up against a wall.
You will not fence the side against the wall. What is the largest area you can enclose?
15.
Find the point on the graph of 𝑦 = 𝑥 2 + 3 that is closest to the point (4, 0).
16.
A metal storage tank is to be constructed in the shape of a cylinder with volume 100 𝑚 3
. What
dimensions would minimize the amount of material required to construct the tank?
Find 𝒇(𝒙) (the most general antiderivative) for each of the following.
17.
𝑓′(𝑥) = 6𝑥 9 − 4𝑥 6 + 13𝑥 3
18.
3+𝑥+𝑥
2
√𝑥
19.
𝑓′(𝑥) = 3√𝑥 + 2 cos 𝑥 20. 𝑓 ′ (𝑥) = 5 sin 𝑥 + 𝑥 −1
Find the antiderivative 𝒇(𝒙) that satisfies the given conditions.
21.
𝑓′(𝑥) = 5𝑥 4 − 2𝑥 5 ; 𝑓(0) = 6
22.
𝑓 ′′ (𝑥) = 24𝑥 3 − 18𝑥 2 + 10𝑥; 𝑓(1) = 4; 𝑓 ′ (0) = 3

A particle is moving with the given data. Find the position of the particle.
23. 𝑣(𝑡) = 𝑡 2 − 5𝑡 + 3; 𝑠(0) = 6
24.
𝑎(𝑡) = cos 𝑡 + 5 sin 𝑡 ; 𝑠(0) = 5; 𝑣(0) = −3