1. (5pts) How many inflection points does the function f (x) = 4x5
5x4 + x + 3 have?
(A) 1
(B) 0
(C) 2
(D) 3
(E) Trick question. f 00 (x) does not exist.
Answer:
2. (5pts) Use Newton’s Method to find x2 , the second approximation to the positive root of
x3
17 = 0,
if the initial approximation x1 = 3.
10
27
5
3 + 17
3 11
25
4
3 + 13
3 29
(A) 3
(B)
(C)
(D)
(E)
Answer:
2
Name:
3. (5pts) A race car sits at the starting line. At time t = 0 seconds, the driver steps on the gas. The
car’s acceleration is given by a(t) = 6t2 feet per second squared. How fast is the car traveling at t = 3?
Hint: Use the fact that velocity is an antiderivative of acceleration.
(A) 54 ft/s
(B) 72 ft/s
(C) 42 ft/s
(D) 50 ft/s
(E) 36 ft/s
Answer:
4. (5pts) By about how much does the function y = tan 1 x change when x goes from 1 to 1.04? Hint:
Use di↵erentials.
(A) .01
(B)
.04
(C) .02
(D) 0
(E)
.03
Answer:
3
5. (10pts) Consider the following function:
f (x) =
1 4
x
4
2x2 + 1
(a) (1pts) Find f 0 (x).
f 0 (x) =
(b) (3pts) Find the critical point(s) of f (x).
(c) (3pts) Use the First Derivative Test to classify the critical points you found in (b) as local
minima, local maxima, or neither.
(d) (1pts) Find f 00 (x).
f 00 (x) =
(e) (1pts) Find the inflection point(s) of f (x).
(f) (2pts) On which interval(s) is f (x) concave down?
6. (5pts) In this problem, we use linear approximation to estimate values.
p
(a) (4pts) Find the linear approximation of f (x) = x at x = 4.
L(x) =
(b) (1pts) Use part (a) to estimate
p
3.9.
p
4
3.9 ⇡
Name:
7. (12 pts) An open cardboard box (with no top) is to be constructed so that the length of the box is
twice its width. The width of the box is labeled w in the picture at the bottom of the page. If the box
is to have a volume of 36 in3 , find the width w and the height h that minimize the amount A (or area)
of cardboard used.
(a) (2pts) Find a formula for A in terms of w and h.
A=
(b) (2pts) Write a constraint equation in terms of w and h that results from the fixed volume.
(c) (2pts) Using (a) and (b) above, write A as a function of w alone.
A=
(d) (6pts) Find the dimensions of the box (specifically w and h) that minimize A.
w=
h=
8. (4pts) Find the value(s) of c guaranteed by the Mean Value Theorem for the function f (x) = x1 on the
interval [1, 4].
c=
5
9. (8pts) Determine the following indefinite integrals:
Z
p
(a)
5x3 sin x + x + 8 sec2 x dx
(b)
Z ✓
6
x
p
3
1
x2
+ 5ex
2
x3
◆
dx
10. (8pts) Compute the following limits:
ex 1
x!0 ln (x + 1)
(i) (4pts) lim
(ii) (4pts) lim
(x
⇡
x! 2 +
⇡
) tan x
2
6