Calculus 11/19/2010
1. Give an ϵ, δ proof for the statement (7)
lim
x→5(4x− 1) = 19
2. Find lim
x→1
(x3− 1)(x + 3)
(x2− 1)(x2+ 4) (7)
3. Let f (x) =
{x2, x < 1
Ax + 4, x≥ 1. Find A given that f is continuous at 1. (7)
4. Let f (x) =
{x2cos(1/x), x̸= 0
0, x = 0. Use the Pinching theorem to show that f′(0) = 0. (7)
5. Find d dx
[
(x2− 4x)−1 d
dx(x + x−1) ]
(8)
6. Find dydx at x = 2 if y = (s + 2)3, s =√
t− 3, t = x2. (8)
7. Find the derivative of (8)
f (x) = sec x cot 3x
8. Evaluate dy/dx at the point (2,−1) if (8)
x2+ 4xy + y3+ 5 = 0.
9. Find the critical points and the local extreme values of (8)
f (x) = (1− x)(1 + x)3
10. Find the largest possible area for a rectangle with base one the x–axis and upper vertices on the
curve y = 8− x2. (8)
11. Sketch the graph of (8)
f (x) = x 1 + x2. Show all critical points, points of inflection, and asymptotes.
12. An object is dropped and hits the ground 10 seconds later. From what height, in feet, was it
dropped? (Hint: g = 32 feet per second per second) (8)
13. A 13–foot ladder is leaning against a vertical wall, forming an angle θ with the ground. If the bottom of the ladder is being pulled away from the wall at the rate of 0.1 feet per second, how fast is the angle θ changing when the top of the ladder is 12 feet above the ground? (8)
