M403K Final Exam (with solutions)

Given December 16, 2000

Exam scores were pretty good.  The class average was a 73; the median was a 77.  However, LOTS of people had trouble with exponentials and logs.  For instance, many people said that ln(x) could never be negative.    (You can't take the log of a negative number, but ln(x)<0 whenever x<1).  Another common mistake was forgetting the rules for combining logs and writing something ln(2 e^2t) = 2(2t), instead of ln(2) + 2t.  A few people also wrote things like e^{-t} + e^{-2t} = e^{-3t}!

Exponentials and logs appear throughout M403L (and throughout mathematics).  This is something you absolutely need to know.

Problem 1.

Consider the function tex2html_wrap_inline55 . Note that this function is only defined for x>0.

a) Find the partition points of f and make a sign chart for f.

Since x is positive, the partition point is at tex2html_wrap_inline65 , that is at x=1. f(x) is positive for x>1 and negative for 0<x<1.  A LOT of people mistakenly said that f(x) was positive for 0<x<1.

b) Find the critical points of f and make a sign chart for f'.

tex2html_wrap_inline79 . This is zero when tex2html_wrap_inline81 , that is at x=1/e. f'(x) is negative for x<1/e and positive for x>1/e.   (A LOT of people said that there were no critical points. This is simply wrong.)

c) Find the inflection points of f and make a sign chart for f''.

f''(x)=1/x is always positive. There are no inflection points.

d) Sketch the curve y=f(x). Mark all important points CLEARLY. [You may find useful the fact that limit tex2html_wrap_inline99 .]

[Sorry, I can't do that online. But the curve starts at (0,0), heads downwards while curving upwards, reaches a minimum at (1/e, -1/e), crosses the x axis at (1,0), and continues to increase (and curve upwards) from then on.]

Problem 2.

Mariposa Ballot Services prints ballots for local governments. They have determined that demand for their ballots does not depend on the quality of their printing, but only on their price, via the relation p = 30 - 5 x, where x is the demand and p is the price. Their cost is C(x) = 10 + 10 x.

a) Compute the revenue function R(x) and the profit P(x).

tex2html_wrap_inline119tex2html_wrap_inline121 .

b) Compute the marginal revenue, the marginal cost, and the marginal profit as a function of x.

R'(x)=30-10x, C'(x)=10, and P'(x)=20-10x.

c) At what production level is the profit maximized? What is the profit at this level?

Where P'(x)=0, that is at x=2. The profit is P(2)=10.

d) At what production level is the revenue maximized?

Where R'(x)=0, namely at x=3. The revenue is R(3)=45.  (You were not required to compute the revenue).

[Language note: Mariposa is Spanish for ``butterfly'']

Problem 3.

A homeowner wants his Christmas display to be as beautiful as possible. He has decided that a beautiful display should have both red and green lights. In fact, he considers the ``beauty index'' to be the product of the number of red and the number of green lights. If red lights cost $2 each and green lights cost $3 each, and if the homeowner has $60 to spend, how many lights of each type should he buy?

Let x be the number of red bulbs, and let y be the number of green bulbs. We have 2x+3y=60, so y=20 - (2/3)x. The beauty index is tex2html_wrap_inline151 .

The maximum is when 0=B'(x) = 20 -4x/3, that is at x=15 and y=10. The homeowner should buy 15 red lights and 10 green lights.

You could just as well let y be the independent variable, and solve for x = (30 - 3y/2), so B= 30y - 3y^2/2.  We then get 0=dB/dy = 30-3y, so y must be 10, and x=15.

Problem 4.

A Geiger counter is used to measure radioctivity. It clicks every time the radiation from a decay hits it. In a particular experiment, it picks clicks at a rate of tex2html_wrap_inline159 .

a) At what time is the counter clicking fastest?

We want to maximize tex2html_wrap_inline161 . We compute tex2html_wrap_inline163 and set it equal to zero, getting tex2html_wrap_inline165 , or tex2html_wrap_inline167 .

b) How many clicks are recorded between time t=0 and time t=3?


Problem 5.

The price p and demand x for a commodity are related by the equation tex2html_wrap_inline179 . In particular, when p=10, x=30.

a) Find dx/dp at the point (p,x)=(10,30).

Take the derivative of the equation with respect to ptex2html_wrap_inline191 , so tex2html_wrap_inline193 .

b) Use the result of part (a) to estimate the demand when the price is 10.1.

Use differentials. p=10, dp=.1, so dx=(dx/dp) dp = -0.5, and tex2html_wrap_inline201 .

Problem 6.

The marginal profit for a company is tex2html_wrap_inline203 . Furthermore, the profit when x=10 is P(10)= 10,000. Find the profit when x=2.

Integrate to get tex2html_wrap_inline211 . Since P(10)=10,000, we must have C=-1000, so P(2) = 2400 -8 -1000 = 1392.

Part II:

Evaluate the following:

a) tex2html_wrap_inline219 , since tex2html_wrap_inline221 grows faster than any other term.

b) tex2html_wrap_inline223 . This is an integration by substitution, with tex2html_wrap_inline225 .

c) tex2html_wrap_inline227 .

d) tex2html_wrap_inline229 .

e) tex2html_wrap_inline231 . Use L'Hopital's rule once to get tex2html_wrap_inline233 .

f) tex2html_wrap_inline235 . This is an integration by substitution with tex2html_wrap_inline237 .

g) tex2html_wrap_inline239 , by the ratio rule.

h) f''(x), where tex2html_wrap_inline243 .

tex2html_wrap_inline245 , and tex2html_wrap_inline247 .

i) tex2html_wrap_inline249 .

j) tex2html_wrap_inline251 . Since tex2html_wrap_inline253 , this is just tex2html_wrap_inline255