M403K First Midterm Exam Solutions. September 21, 2000

1. Consider the function

a) Find *f*'(*x*).

Soln:

b) Find the slope of the line tangent to the curve *y*=*f*(*x*)
at
*x*=0.

Soln: Slope equals

c) Find the equation of this tangent line.

Soln: Since *f*(0)=1, the tangent line is *y*-1 = 2.4(*x*-0),
or
*y* = 2.4 *x* + 1.

d) Use this tangent line to approximate *f*(0.01).

Soln: So *f*(0.01) is very close to 2.4(0.01) + 1 = 1.024.

2. From the following table, estimate *f*'(1). Indicate clearly
how you obtain your answer:

Solution: For *h* ranging from -.03 to +0.03, look at [*f*(1+*h*)-*f*(1)]/*h*:

As *h* approaches zero from either side, [*f*(1+*h*)-*f*(1)]/*h*
approaches 3, so *f*'(1) equals 3.

3. Evaluate the following limits, if they exist (or write DNE if they do not).

a)

As , the numerator goes to zero while the denominator goes to -3, so the ratio goes to zero.

b)

As
both the numerator and denominator go to zero, so we have to be more careful.
The numerator equals (*x*-2)(*x*+1). Canceling a factor of (*x*-2)
from numerator and denominator we have

c)

As
, the numerator goes to 1 while the denominator goes to zero *from above*,
so the limit is
. Put another way, if
*x* is slightly bigger than 1 (say 1.001), then
the numerator is close to 1 while the denominator is a small possitive
number (like .001), so the ratio is a large positive number (like 1000).

d)

As
, the numerator goes to 6, but the denominator goes to zero. If *x*
is slightly bigger than 4, then the denominator is a small positive number
and the ratio is a large positive number. That is,
. But if *x* is slightly less than 4, then the denominator is a small
negative number and the ratio is a large negative number. That is,
. Since the limits from the two sides do not agree, there is no overall
limit. The correct answer is ``DNE''.

4. Take the derivatives of the following functions with respect to *x*.
You do not need to simplify your answers:

a)

Use the chain rule: .

b)

Use the product rule: .

c)

You can either use the quotient rule or the chain rule. Since , .

d)

Use the quotient rule. .

5. New England Widget Technologies (NEWT) makes expensive high-tech
widgets. Their marketing department has determined that the demand function
is *x* = 4000 - 2*p*, where *x* is the number of widgets
sold and *p* is the price. Their cost function is *C*(*x*)=
200,000 + 1000*x*.

a) Find the price *p*(*x*) and the revenue *R*(*x*)
as a function of *x*.

Soln: *p*(*x*) = (4000-*x*)/2 = 2000 - *x*/2.
.

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

Soln: Marginal cost = *C*'(*x*) = 1000. Marginal revenue =
*R*'(*x*)=2000-*x*. Marginal profit = *P*'(*x*)
= *R*'(*x*)-*C*'(*x*) = 1000 - *x*.

c) The company has a current production level of *x*=1500. To increase
*revenue*, should the company increase or decrease production? [Note:
you do *not* need to compute the optimal level of production. You
just need to say whether it is higher or lower than 1500.]

Soln: Since *R*'(1500)=2000-1500=500, each additional widget produced
will increase revenue by $500. So to increase revenue the company should
increase production.

c) The company has a current production level of *x*=1500. To increase
*profit*, should the company increase or decrease production?

Soln: Since *P*'(1500)=1000-1500=-500, each additional widget made
will
*decrease* profits by $500. This makes sense, because each additional
widget costs $1000 to make and only brings in $500 in revenue. So, to increase
profit, the company should decrease production.