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, x4=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.      

Note:  the drawing got changed very slightly between the original on the test and here.  The answers should all be the same, but if you notice minute differences that is why.

 

 

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)