1. (7 Points) Suppose that y = f (x) is a differentiable function on (−∞, ∞) so that
π
π
f (tan x) = x, − ≤ x ≤ .
2
2
0
Find f (1).
By Chain rule, f 0 (tan x) sec2 x = 1. If tan x = 1 for −π/2 ≤ x ≤ π/2, x = π/4.
Then sec2 x = tan2 x + 1 = 2. We find f 0 (1) = 1/2.
2. (8 Points) Show that x − ln x > 1 for x > 1.
Let f (x) = x − ln x. Then f 0 (x) = 1 − 1/x > 0 for x > 1. We find f is increasing
and hence f (x) > f (1) = 1.
3. (10 Points) Use implicit differentiation to find the tangent line at (−1, 0) of the graph
of the function 6x2 + 3xy + 2y 2 + 17y = 6. (8 points for the slope y 0 |(x,y)=(−1,0) and
2 points for the tangent line.)
See homework solution. 1
2x + 1
−
.
4. (10 Points) Compute lim
x→0+
x
sin x
Ans: 2.
5. (30 Points) Compute
the following indefinite integrals
Z
(1) (8 Points)
sec xdx.
See in classZ note.
(2) (8 Points) (3x2 − 1) ln xdx.
We know d(x3 − x) = (3x2 − 1)dx. Hence
Z
Z
2
(3x − 1) ln xdx = ln xd(x3 − x)
Z
1
3
= (x − x) ln x − (x3 − x) · dx
x
Z
= (x3 − x) ln x − (x2 − 1)dx
x3
− x + C.
3
a. (6 Points) Find constants A, B, C such that
= (x3 − x) ln x −
(3)
x4 + x2 + x − 1
A Bx + C
=x+ + 2
.
3
x +x
x
x +1
Z 4
x + x2 + x − 1
b. (8 Points) Compute the integral
dx.
x3 + x
1
2
See in class note.
x2 + 1
for x 6= 0.
x
(a) (2 Points) Find all of the vertical asymptotes of y = f (x).
(b) (2 Points) Find all of the oblique asymptotes of y = f (x).
(c) (2 Points) Compute f 0 (x).
(d) (2 Points) Find the critical points of y = f (x).
(e) (2 Points) Find the local maximum and the local minimum of y = f (x).
(f) (3 Points) Identify the intervals on which the function are increasing and decreasing.
(g) (2 Points) Compute f 00 (x).
(h) (2 Points) Identify the intervals on which the function are concave up and
concave down.
(i) (3 Points) Sketch the graph.
See in class note.
√
7. (15 Points) A right triangle whose hypotenuse is 3m long is revolved about one
of its leg to generate a right circular cone.
6. (20 Points) Let f (x) =
(1) (5 Points) Let x be the radius of the cone. Find the volume of the cone in
terms of x.
(2) (10 Points) Find the greatest volume that can be made this way.
8. Bonus (10 Points) Let y = f (x) be a function continuous on [0, 1] and differentiable
on (0, 1). Suppose that f (0) = f (1) = 0. Show that there exists c ∈ (0, 1) so that
f 0 (c) + f (c) = 0.
Let g(x) = ex f (x). Since f (0) = f (1) = 0, g(0) = g(1) = 0. By Rolle’s theorem,
there is 0 < c < 1 so that g 0 (c) = 0. Since g 0 (x) = ex (f 0 (x) + f (x)), if g 0 (c) = 0,
then by ec > 0, f 0 (c) + f (c) = 0.