**1. Related rates.**

Consider the curve , .

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 *x*:
, so
.

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)

b) .

c)

d) 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?

*p*=1.

**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?

. 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?

inches.

**Problem 5. Indefinite integrals.**

Evaluate the following integrals:

a)

b) . (Integrate by substitution with .)

c) . (Integrate by substitution with .)

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

**Problem 6. Area under a curve.**

We are interested (OK, OK, your instructor is interested) in finding
the area under the curve
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
(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.**

and *a*=1, so
. Thus
, and our estimated area,
, works out to