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