Final Exam solutions from 1991

This exam consists of 2 parts. Part I contains 10 short ``grab-bag'' questions, each worth 4 points. Part II contains 5 longer problems, each worth 12 points. The exam was closed-book, but allowed a crib sheet. Unlike this year, a non-graphics calculator was allowed.

Evaluate the following derivatives, integrals, and limits, and simplify the following expressions involving exponentials and logs.

For problems A-D, you do NOT have to reduce your answers to the simplest possible forms. Any correct answers will do.

However, for problems E-J, I do want answers in the simplest possible form. For example, if the answer is zero, then writing ``ln(1)'' (which DOES equal zero, of course) will only get you partial credit. Also, you should show your reasoning. Just pulling numbers from a calculator will get you NO credit.

Some of the limits may not exist. If they don't exist, write ``DNE'' or ``does not exist'' in the box.

A. Find f'(x), where tex2html_wrap_inline41 .

Use the product rule. tex2html_wrap_inline43

B. Evaluate: tex2html_wrap_inline45 .

tex2html_wrap_inline47 .

C. Find f'(x), where tex2html_wrap_inline51

Product rule again: tex2html_wrap_inline53 .

D. Find f'(x), where tex2html_wrap_inline57 and tex2html_wrap_inline59 .

Chain rule. tex2html_wrap_inline61 .

E. Evaluate (and simplify): tex2html_wrap_inline63

Integrate by substitution with tex2html_wrap_inline65tex2html_wrap_inline67 . Evaluating at x=2 and x=0 gives tex2html_wrap_inline73 .

F. Evaluate (and simplify): tex2html_wrap_inline75

By L'Hopital's rule, this equals tex2html_wrap_inline77 .

G. Evaluate (and simplify): tex2html_wrap_inline79

By L'Hopital's rule, this gives tex2html_wrap_inline81 .

H. Simplify: tex2html_wrap_inline83

tex2html_wrap_inline85 , so tex2html_wrap_inline87 .

I. Simplify: tex2html_wrap_inline89 .

J. Find dy/dx at the point (2,3), where y is given by tex2html_wrap_inline95 . (Your answer should be a NUMBER, not something like tex2html_wrap_inline97 .)

tex2html_wrap_inline99 , so tex2html_wrap_inline101 .

This part consists of 5 longer problems, each worth 12 points.

Problem 1. A driver goes out on a 4-hour car trip. v(t), his velocity at time t, is given by the formula:


Here t is measured in hours, and v is measured in miles/hour.

Note that parts (a) and (b) below can be done independently of each other. Part (a) involves differentiation, while part (b) involves integration.

a) At what time is the driver driving the fastest? How fast is he going at that time?

To find the critical points, take the derivative and set it equal to zero: 0 = v'(t) = 30 - 15 t, so t=2. By the second derivative test, this is a local max. It is also a global max. So he is going fastest at t=2 and his speed at that time is v(2) = 40 + 60 - 30 = 70.

b) How far does the driver travel in 4 hours?

tex2html_wrap_inline121 miles.

Problem 2. Consider the function tex2html_wrap_inline123 .

a) Find all the local maxima of f(x), all the local minima, and all the points of inflection.

tex2html_wrap_inline127 , so the critical points are at x=-1, x=0 and x=+1. tex2html_wrap_inline135 , so f''(-1);SPMgt;0, f''(0);SPMlt;0 and f''(1);SPMgt;0. Therefore -1 and 1 are local minima, while 0 is a local maximum. The points of inflection are where f''(x)=0, namely at tex2html_wrap_inline151 .

b) For what values of x is f increasing?

For -1 < x < 0 and for x > 1.

c) For what values of x is the graph of f concave up?

For tex2html_wrap_inline165 and for tex2html_wrap_inline167 .

Problem 3. C-14, an isotope of carbon, is radioactive. Suppose the radiation from a sample of C-14 decreases at a rate of 0.014% per year, compounded continuously (this figure is pretty accurate).

a) What is the half-life of C-14 (in other words, how long do you have to wait for half the radioactivity to be gone)?

Half life = tex2html_wrap_inline169 years.

b) If 1 pound of C-14 were deposited 2000 years ago, how much of it would be left today?


c) If a sample of ancient wood contains only one tenth of the C-14 that it used to, how old is it?

tex2html_wrap_inline173 , so tex2html_wrap_inline175 , so tex2html_wrap_inline177 , so tex2html_wrap_inline179 16,500 years.

4. a) Use differentials to estimate tex2html_wrap_inline181 .

tex2html_wrap_inline183 , x=10, dx=0.01, tex2html_wrap_inline189tex2html_wrap_inline191 .

b) The radius of a ball is increasing at a rate of 1 ft/sec. At what rate is the volume increasing when the radius reaches 10 feet? (Hint: the volume of a sphere of radius r is given by tex2html_wrap_inline195 ).

tex2html_wrap_inline197 cubic feet/second.

5. Consider the function


a) Does tex2html_wrap_inline201 exist? Why or why not? If it exists, what does it equal?

By L'Hopital's rule, the limit is tex2html_wrap_inline203 .

b) Is f(x) continuous at x=1? Why or why not?

Since the limit equals the value of the function, the function is continuous at x=1.