Math 160                                                                             Name:                                  Key

Test 1                                                                                    October 1, 2013

You must SHOW ALL WORK to receive credit

Include units where appropriate

All numerical answers should be exact or rounded to at least 3 decimal places unless otherwise indicated.

Note  -answers are shown, but not all work is provided – make sure you can supply the missing steps!

1.        Let

a.        Evaluate f(2)

b.      Evaluate and simplify  f(2+h)

2.       Let

a.       Evaluate f(g(4))

b.      Write a formula for the function g(f(x))

c.    What is the domain of the function g(f(x))?

3.       A house was worth \$250,000 when it was new.  20 years later its value is \$450,000.

Assume that the value of the house is a linear function of time.

a.       Find a formula for the value of the house as a function of t, the number of years since it was built.

b.      INTERPRET in a complete English sentence what the slope of this line means in practical terms.

The value of this house increased at a rate of \$10,000 per year during this 20 year span. (Or, the value increased by \$10000 each year)

4.       Consider again the house from #3.  Its value was \$250,000 when new and \$450,000 after 20 years.

Assume now that the value of the house is an exponential function of time.

Find a formula for the value of the house as a function of t, the number of years since it was built.

5.       Evaluate the following limits using the method indicated:

 h (2^(3+h)-8)/h 0.1 5.7419 0.01 5.5644 0.0002 5.5456 0.00001 5.5452 -0.0003 5.5446 -0.000001 5.5452

Based on the values in this table, it appears that the answer is approximately 5.545

1

(remember – what you need is the value of the function when x is near 3, but not equal to it)

6.       Use the definition of the derivative  (not a shortcut formula) to find

All questions on this page refer to the function y=f(x) shown in the graph.

7.       Carefully draw the tangent line to

this curve at x=2 (draw it on the graph)

(shown in red)

8.       Approximate the following values.

a.  f(2)

6

-1/2  - this is the slope of the tangent line as drawn, and the derivative means exactly that.

9.   a. Determine the interval(s) on which f’(x) is positive.  Justify your answer!

f’(x) is positive when f(x) is increasing.  This occurs on the intervals (approximately)

(0.1 , 1.7) U (6.1, 8.8)

b.  Determine the interval(s) on which f’’(x) is positive.  Justify your answer!

f’’(x) is positive when f is concave up.  This is approximately on the interval (3.5,7.5)

10.  What is the average rate of change of f(x) on the interval [0, 4]?

** The question added to this sheet on the take home quiz asked for the equation of the tangent line found above.  It will obviously depend on exactly how you drew the line, but as I drew it, the line had a slope of -1/2 and went through the point (1,6), so it was