Math 181 Name: __ Key __

Test #1 June
20, 2013

__You must SHOW ALL WORK to receive credit__

Include units where
appropriate

Problems asking you
to INTEPRET or EXPLAIN should be answered __in complete English sentences__.

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

1. Let m(x) = ln(x-4) and n(x) = ^{ }2x^{ }. Find each of the following:

a.

b.

*For
m(x), y=ln(x-4), so for m ^{-1}(x), x=ln(y-4). Solving for
y:*

c. Express m(x) as a composition , where f and g are two simpler functions.

2. a. Carefully prove that the equationmust have a real root in the interval (0, 2).

*Since f is a polynomial it is continuous
everywhere and so the Intermediate Value Theorem applies. In particular, f must take on every value
between -1 and 13 somewhere in the interval (0,2). So f(x)=0 somewhere
in this interval. Where f(x) =0, x ^{4}=2x+1,
as desired.*

b. Use your graphing calculator to
approximate all solutions to . No work needs to be
shown for this part.

*There are 2 solutions, where the graphs of
these 2 functions cross. They occur at x* = -.475
and x = 1.395

3. A house was worth $250,000 when it
was new. Now, 20 years later, it is
worth $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 the number of years since it was built.

*The line connects the points (0,250000) and (20, 450000).
This has slope ** and so the line is y=10000t + 250000, where t
is the time in years, and y is the value in dollars.*

b) *INTERPRET*
what the slope of the line from part (a) means __in practical terms__.

*The slope of 10000 indicates that the value
of the house has gone up by $10000 each year since the house was built.*

4. A house was worth $250,000 when it was
new. Now, 20 years later, it is worth
$450,000. Assume this time 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 the number of years since it was built.

*We need the exponential curve through the
same two points as in the previous problem.
There are several possible forms for the answer – here is one:*

5. Evaluate the following limits if they
exist:

y=f(x)

6. Evaluate
the following limits if they exist, or explain why they do not exist. Briefly justify each answer - no credit will
be given for unsupported answers!

*DNE, since the 2 1-sided limits at x=2 do not
agree.*

*1.
** The
closer x gets to 3 the closer y gets to 1 (what happens *

*when** x actually
equals 3 is not relevant)*

*By the limit laws, we can split this up as ** as long as each of these limits exists. The first limit is 2 (by the graph) and the
second is 1 (by the limit laws), so the answer is 3*

7. __Use the
definition of the derivative__ (either form, but not the shortcuts) to find if .

*At x=2, this has the value 3, so f’(2) = 3.*

*Note:
You could also have directly evaluated
*

8. The graph of y=f(x) is shown:

Carefully draw the
tangent line to this graph at x=2.

*(see in red)*

9. For this function f(x), approximate the
following values (some may not be possible to determine exactly, but a
reasonable estimate, sufficiently justified, will suffice)

*Approximately 6.3 – the height of the curve
at x=2*

b.

*Approximately -.5 (or slightly
more negative –anything in the -.5 to -.75 range could be acceptable if agreed with
your drawing) *

c. The interval(s) on which is
positive

*This is where the curve is increasing, which is on the intervals
(-∞,1.7) and (6,8.8)*

d. The interval(s) on which is positive

*This is where the curve is concave up, which is the interval (4,8)*