Math 181

Test #2 Name: __ key __

July 9, 2013

As always, you must **SHOW ALL WORK** to receive credit

Include units in
answers wherever appropriate

Problems that ask
for interpretations should be answered __in complete sentences__, in the
context of the problem

1. The height of a
model rocket *t* seconds after it is
launched is .

a. What is the velocity of the rocket at t=2
seconds?

b. What is the acceleration of the rocket
at t=2 seconds?

c. What
is the average velocity of the rocket over the first 2 seconds of its flight?

2. Let H(t) be the
temperature, in °F, of a cup of coffee *t*
minutes after it is set on the table.

a. What are the units of H'(t)?

b. Suppose H'(10) = -3. Interpret what this
means in practical terms.

10 minutes after the
coffee is set on the table, its temperature is decreasing at an (instantaneous)
rate of 3 degrees Fahrenheit per minute.

*For questions 3
and 4 your answers must be justified by calculations, not by your graphing calculator.*

3. Let .
Determine with justification the __exact__ interval(s) on which f(x)
is increasing.

. The
denominator is always positive, so f’(s) is positive whenever the numerator is,
which is when x>-½ . Since f(x) increases when f’(x) is positive,
this is the answer.

4. Let . Determine,
with justification, the __exact__ interval(s) on which g(x) is concave up.

G is concave up when
g’’ is positive, which is when x<-5 or x>1

5. ACME’s cost of producing x widgets per week
is given by

a. Evaluate , and interpret this quantity in the context
of this application.

*It costs ACME $1400 per week to make 100
widgets.*

b. Evaluate , and interpret this quantity in the context
of this application.

*When ACME is making 100 widgets per week, its
marginal cost is $4/widget, or in other words each additional widget will costs
an additional $4.*

6. Earlier this
semester we modeled the number of hours of daylight in Rockville *t *days after the beginning of the year with
the function

a. Find a formula for *f*
'(t).

b. Find the value of * **f* '(190) (round your answer to the nearest
thousandth). Then __interpret__ this
quantity in the context of the application.

190 days into the
year (ie, today!), the length of the days is decreasing
by about .0164 hours (about 1 minute) each day.

7. Use implicit differentiation to find the
derivative at (2,-1) of the curve defined by

At (2,-1), this
becomes

8. According to Wolframalpha,
the radius of the earth is 6368 km. Use
differentials to estimate the error in approximating the surface area of the
earth if you use 6371 km as the radius.
(This latter quantity, by the way, is the radius according to
Google). *Note: the surface area of a
sphere of radius r is *

9. Use a linearization of the function at an appropriately chosen point to approximate the value of

*The closest point to x = 10 where we can
easily and precisely find the linearization is at x=8:*

*The linearization (tangent line) is therefore **. At x=10, we get*