Sample Final, Math 1251, Spring 1997
Calculators allowed. Open book. 3 hours.
Suggestion: Be sure that you know how to differentiate!
1. (a) Suppose that y=f(x) is polynomial of degree 17. What is the 20th derivative of f(x)?
(b) Find dy/dx if y=x^2*e^(-\sqrt(x)/4), y=sin(3x)/x, y^2*x+x^4*e^y=12.
2. Compute the following limits.
(a) \lim_{h\to0} {(ln(x+h)-ln(x))/h}.
(b) \lim_{x\to\infty} {e^(-3x)+x^4-e^x}/ {x^78-4e^x}.
3. True or false. If true, give a reason. If false, give a counterexample.
(a) If f(0)=2 and f'(x)>0 for all x, than f(x)>2 for any x>0.
(b) If f is differentiable at 1, then f is continuous at 1.
(c) If f'(0)=f'(1)=0, then x=0 is a local min for f(x).
(d) If lim_{x\to 0} (f(x)+g(x)) exists, than lim_{x\to 0} f(x) also exists.
(e) The function f(x)=(x-2)^4 on the interval [0,3] is a 1-1 function.
(f) The function f(x)=e^(-4x) is concave upward on (-\infty, \infty).
4. Let f(x)=x+ln(x) and let g=f^{-1}. Find f(e), g(e+1), f'(e), and g'(e+1).
5. A boat is traveling on a straight line, always moving away from a dock. Its distance from the dock at time t is \pi*t+t^5/7.
(a) Find the average velocity of the boat during the first two hours of travel ( i.e., between time t=0 and t=2).
(b) Find the instantaneous velocity at time t=4.
6. Sketch the graph of a function f such that
f'(x)>0 for x all real values of x.
7. Choose 7A or 7B. Do not do both.
7A. Find the absolute maximum and absolute minimum values of f(x)=x^4/4-2x^3/3-5x^2/2+6x on the interval [-3,2]. Show your work, or you will receive no credit. Also find all local max/mins inside [-3,2], and prove that they really are local max/mins.
7B. Define \lim_{x\to\infty}f(x)=3 intuitively, and then rigorously.
8. Gravel is being dumped from a conveyor belt at a rate of 30 ft^3/min. Its coarseness is such that it forms a conical pile whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 feet high?
9. A right circular cylinder is inscribed in a cone with fixed height h and fixed base radius r. Find the largest possible volume of such a cylinder.