1. (7 Points) Suppose that y = f (x) is a differentiable function on (−∞, ∞) so that f (tan x) = x, −π
2 ≤ x ≤ π 2. Find f0(1).
By Chain rule, f0(tan x) sec2x = 1. If tan x = 1 for −π/2 ≤ x ≤ π/2, x = π/4.
Then sec2x = tan2x + 1 = 2. We find f0(1) = 1/2.
2. (8 Points) Show that x − ln x > 1 for x > 1.
Let f (x) = x − ln x. Then f0(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 + 2y2+ 17y = 6. (8 points for the slope y0|(x,y)=(−1,0) and 2 points for the tangent line.)
See homework solution.
4. (10 Points) Compute lim
x→0+
2x + 1
x − 1
sin x
. Ans: 2.
5. (30 Points) Compute the following indefinite integrals (1) (8 Points)
Z
sec xdx.
See in class note.
(2) (8 Points) Z
(3x2− 1) ln xdx.
We know d(x3− x) = (3x2− 1)dx. Hence Z
(3x2− 1) ln xdx = Z
ln xd(x3− x)
= (x3− x) ln x − Z
(x3− x) · 1 xdx
= (x3− x) ln x − Z
(x2− 1)dx
= (x3− x) ln x −x3
3 − x + C.
(3) a. (6 Points) Find constants A, B, C such that x4+ x2+ x − 1
x3+ x = x +A
x +Bx + C x2+ 1 . b. (8 Points) Compute the integral
Z x4+ x2+ x − 1 x3+ x dx.
1
2
See in class note.
6. (20 Points) Let f (x) = x2+ 1
x for x 6= 0.
(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 f0(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 de- creasing.
(g) (2 Points) Compute f00(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.
(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
f0(c) + f (c) = 0.
Let g(x) = exf (x). Since f (0) = f (1) = 0, g(0) = g(1) = 0. By Rolle’s theorem, there is 0 < c < 1 so that g0(c) = 0. Since g0(x) = ex(f0(x) + f (x)), if g0(c) = 0, then by ec> 0, f0(c) + f (c) = 0.
