Sample Final, Math 1251, Spring 1997, written by Scot Adams
Calculators allowed. Open book. 3 hours.
Suggestion: Be sure that you know how to differentiate!
1. Let L be a function and assume that the graph of y=L(x) is a line. Compute L''(7).
2. Compute the following limits.
(a) \lim_{h\to0} {((x+h)e^{x+h}-xe^x)/h}.
(b) \lim_{x\to-\infty} {x^4+3x-7+sin(e^{x^2})}/ {e^x+7x^4-cos(e^{-x})}.
3. True or false. If true, give a reason. If false, give a counterexample.
(a) No rational function has more than one horizontal asymptote.
(b) If f is continuous at 1, then f is differentiable at 1.
(c) No two functions have the same derivative.
(d) Any odd degree polynomial has a root.
(e) The function f(x)=x+e^x is one-to-one.
4. Let f(x)=x+e^x and let g=f^{-1}. Compute g^{-1}(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 t^3+t^5.
(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\in(-2,2)
f'(x)<0 for x\in(-3,-2)\cup(2,3)
f'(x)=0 for x\in(-4,-3)\cup(3,4)
f'(x)=0 for x=-4,x=-3,x=-2,x=2,x=3,x=4
f(x)>0 for x\in(-4,-3)
f(x)<0 for x\in(3,4).
7. Choose 7A or 7B. Do not do both.
7A. Find the absolute maximum and absolute minimum values of f(x)=x^3+9x^2+6x+7 on the interval [-3,0]. Show your work, or you will receive no credit.
7B. Define \lim_{x\to\infty}f(x)=3 intuitively, and then rigorously.
8. A balloon is released 500 feet away from an observer. If the balloon rises vertically at the rate of 100 feet per minute and at the same time the wind is carrying it horizontally away from the observer at the rate of 75 feet per minute, at what rate is the angle of inclination of the oberserver's line of sight changing 6 minutes after the balloon has been released?
9. Two hallways, one 8 feet wide and the other 6 feet wide, meet at right angles. Determine the length of the longest ladder that can be carried horizontally from one hallway into the other.