Math 12—Precalculus—Chapter 2 Test Solutions

 

1.      Consider the following equations relating x and y :

a.                           y is a function of x, but x is not a function of y.	b.                Neither y nor x is a function of the other variable.
c.    y is a function of x, but x is not a function of y.

 

2.      Let  and   

a.       What is the domain of  f  ? 
ANS:  The domain of f is where  or

b.      What is the domain of  g ?
ANS:  The domain of g is where , which includes all real numbers or

c.       ANS: 

d.      What is the domain of  ?
ANS:  The domain of  is the subset of the domain of g such that the range of g is also in the domain of f.  Now, the range of g can be found by solving for y in terms of x:
 Viewing this last relation as quadratic in x, we require that its discriminant is non-negative:
that is, with  we require that , so , which is in the domain of f, so the domain of  is .
To illustrate, on the TI85, we can define the composition as y2  as shown in the first screen below, then capture the function’s curve over -40<x<40.  You can see that as y1 approaches
-1.4, y2 approaches (asymptotically to the left) -4.4/sqrt(7) ~ -1.66 and as y1 approaches 1.4,  y2 approaches (asymptotically to the right) -1.6/sqrt(7) ~ -0.6.  Thus the function has two horizontal asymptotes, one to the left and one to the right, as shown below.

 

3.      Suppose the rate, R, of population growth is jointly proportional to the present population size, p,  and the amount by which that size falls short of the carrying capacity C – p.

a.       Assuming the constant of proportionality is k, write an equation relating R to p.  This equation will also involve the parameters C and k.
ANS: 

b.      Suppose the carrying capacity is C = 1000 and a polution of p = 800 yields a population growth rate R = 3%.  What is the value of k?  What will the growth rate be if p = 500?
ANS:  Plugging in, we get , thus a population of 500 will produce a growth rate of

4.      Find the average rate of change of the function   over the interval .
ANS:  .

5.      Consider the function .

a.       Make a table of values and sketch a graph of this function.

ANS:  Since this is an even function, +/- inputs correspond to the same output:

b.      The table for   can be found by adding 10 to all the x values and subtracting the corresponding y values from 100 (at right).  Note in the graph that the curve is flipped vertically, the horizontal asymptote asymptote is shifted up 100 and the line of symmetry is shifted 10 to the right.  The graph is shown together with the graph of f above.

6.      Consider

a.          Write the function in vertex form.
ANS: 

b.      Sketch a graph for the function showing the coordinates of the vertex and all intercepts.

c.       ANS:  The intercepts are at  and (0,8) while the vertex is at .    Note that this graph would be obtained from  by reflecting in the x-axis, shifting up 8.125 and shifting left ½, so that .

7.      Find the inverse function for  and sketch a graph of f and its inverse together, showing the symmetry through the line y = x.

ANS:  The following equations are equivalent: Thus, The graphs are at right, scaled to show symmetry through the line y =  x.

ANS:  Let h be the height and r the radius of the cylinder.  Then r +  h = 5 and so h = 5 – r.  Substituting into the volume formula, , which has a local max near  (4/3,58.1776), as shown:

8.      Suppose the sum of the height of a circular cylinder with the radius of its base is 5 units.  What is its maximum volume?