Math 4111
Test 3
Spring 1996

No books, notes etc. are permitted.
Show all your work! Box in your answers!
The test has 7 problems on 4 pages.
Read the problems carefully!

Problem 1 (25 points)   Compute the derivative of the following functions. You need not simplify your answers!

1.

$f(x)=x^5-3x^3+2x^2-\sqrt{2}x+2$

2.

$$f(x)=[\sin(x)]^2 \cdot \cos(x)$$

3.

$$f(x)=\frac{x^2 \cos(2x)}{x^3+e^{-x}}$$

4.

$$f(x)=\log_4(x)$$

5.

$$f(x)=\arcsin(\frac{1}{x^2})$$

Problem 2 (15 points)  

1.

Find the first three derivatives of $$f(x)=2^{(2x-3)}$$.

2.

Find the 100th derivative of $$f(x)=2^{(2x-3)}$$.

Problem 3 (10 points)   Find an equation for the tangent line of $$g(x)=\frac{x^2+5}{x-5}$$ at x=0.

Problem 4 (10 points)   Let $$B(t)=P(t)\cdot(1.05)^{20-t}$$. Given that P(8)=120 and P'(8)=6, find B'(8).

Problem 5 (15 points)   A 300 feet tall hotel has an elevator on the outside of the building. You are standing by a window 100 feet above the ground and 150 feet away from the hotel, and the elevator descends from the top of the hotel at a constant speed of 20 feet per second, starting at time t=0, where t is measured in seconds. Let $\theta$ be the angle between the line of your horizon and your line of sight of the elevator.

1.

Find a formula for h(t), the elevator's height above ground, as it descends from the top of the hotel.

2.

Express $\theta$ as a function of time t.

3.

If the rate of change of $\theta$ with respect to time t is a measure of how fast the elevator appears to you to be moving, at what height will the elevator be when it appears to be moving the fastest?

Problem 6 (15 points)   Find the quantity $$\frac{dy}{dx}$$ at the point (8,8) on the curve

x^2/3+y^2/3=8.

Problem 7 (10 points)   A spherical cell is growing at a constant rate of $250 \mu m^3$ per day. At what rate is its radius increasing when the radius r is $10 \mu m$? (The volume V of a sphere with radius r is given by $$V=\frac{4}{3}\pi r^3$$.)