M403K Third Midterm Exam Solutions
Exam given April 11, 2002

1. Related rates.

Consider the curve tex2html_wrap_inline63tex2html_wrap_inline65 .

a) Find the slope of the line that is tangent to the curve at the point (2,3).

Take the derivative of the equation with respect to xtex2html_wrap_inline69 , so tex2html_wrap_inline71 .

b) A particle is moving along the curve. Its x-coordinate is increasing at a rate of 10 units/second. How fast is y changing when (x,y)=(2,3)?

There are two reasonably easy solutions. One is to use the result from (a): dy/dt = (dy/dx)(dx/dt) = 2(10) = 20 units/second.

The other method is to start from scratch, and take the derivative of the equation with respect to t:


Plugging in values of x, y and dx/dt gives 6 (dy/dt)=120, so dy/dt = 20, as before.

Problem 2. L'Hopital's Rule Evaluate the following limits:

a) tex2html_wrap_inline95

b) tex2html_wrap_inline97 .

c) tex2html_wrap_inline99

d) tex2html_wrap_inline101 L'Hopital's rule does not apply here.

Problem 3. Elasticity of Demand

The demand x for a new toy depends on its price p via the demand equation


a) Compute the elasticity of demand E(p) as a function of p.


b) For what values of p is the demand elastic? For what values of p is the demand inelastic?

When p>1, E<-1 and the system is elastic. [Under these circumstances we should lower the price to increase revenue.]

When p<1, E>-1 and the system is inelastic. [To raise revenue, raise the price].

c) What value of p will maximize revenue?


Problem 4. Horse sense

For the first two years of life, a pony's height H(t) grows at a rate


(where height is measured in inches and time in years). At age 1, the pony is 45 inches tall.

a) How tall was the pony at birth?

tex2html_wrap_inline135 . To evaluate the constant, use the fact that H(1)=45, so 45 = 15 - 1 + C, so C=31. Now plug back in to get


So when t was zero, H was 31.

b) How tall will the pony be at age 2?

tex2html_wrap_inline149 inches.

Problem 5. Indefinite integrals.

Evaluate the following integrals:

a) tex2html_wrap_inline151

b) tex2html_wrap_inline153 . (Integrate by substitution with tex2html_wrap_inline155 .)

c) tex2html_wrap_inline157 . (Integrate by substitution with tex2html_wrap_inline159 .)

d) tex2html_wrap_inline161 . (Integrate by substitution with u=2x+1.)

Problem 6. Area under a curve.

We are interested (OK, OK, your instructor is interested) in finding the area under the curve tex2html_wrap_inline165 between x=1 and x=4.

a) Estimate this area using 3 rectangles. Your final answer should be an explicit number, like 13 or 152.

Each rectangle has width (4-1)/3 = 1. The three rectangles have height f(2), f(3) and f(4), so the estimated total area is f(2)+f(3)+f(4) = 9+19+33 = 61. [If you used the function values at 1, 2 and 3 instead of 2, 3, and 4, I gave full credit. The answer then would be 31]

b) Estimate the area using N rectangles. You can leave your answer as a sum, like tex2html_wrap_inline181 (no, that's not the right answer). Everything in the sum needs to be clearly defined, but YOU DO NOT NEED TO SIMPLIFY OR EVALUATE THE SUM.

tex2html_wrap_inline183 and a=1, so tex2html_wrap_inline187 . Thus tex2html_wrap_inline189 , and our estimated area, tex2html_wrap_inline191 , works out to