Chapter 4 Homework

1.  Given the equation

$f\left(x\right)=\frac{1}{3}{x}^3–x^2+x–6$

find the interval where the function f is increasing.

2. Use the curve sketching guideto sketch the graph of the function

$f\left(x\right) = \frac{x–2}{x+2}$

3.  Find all possible Vertical Asymptote(s) of the graph of the function

$f\left(x\right)=\frac{5x}{x^2–2x–8}$

4.    Find all possible Vertical Asymptote(s) of the graph of the function

$f\left(x\right)=\frac{x^2+x}{x(x–1)}$

5.  Find the absolute maximum value, if any, of the function

$h\left(t\right) =t^3–6t^2$

on [2,5]

6.  Find the absolute minimum value, if any, of the function

$$\begin{gathered} f\left(x\right)=\frac{x^2}{x–1} \\ on [–1,3] \end{gathered}$$

7.   Odyssey Travel Agency’s monthly profit (in thousands of dollars) depends on the amount of money each x (in thousands of dollars)

spent on advertising each month according to the rule

$P\left(x\right)=–x^2+8x+20$

To maximize its monthly profits, what should be Odyssey’s monthly advertising budget?

8.  The weekly demand for DVDs manufactured by Herald Media Corporation is given by

$p=–0.0005x^2+60$

where p denotes the unit price in dollars and x denotes the quantity demanded.

The weekly total cost function associated with producing these discs is given by

$C\left(x\right) = –0.001x^2+18x+4000$

where C(x) denotes the total cost (in dollars) incurred in pressing x discs. 

Find the production level that will yield a maximum profit for the manufacturer.

Hint : Use the quadratic formula

9.  The total monthly cost (in dollars) incurred by Carlota Music in manufacturing x units of its Professional Series guitars is given by the function 

$C\left(x\right) = 0.001x^2+100x+4000$

Determine the production level that will result in the smallest average production cost.

10.  A man wishes to construct a cylindrical barrel with a capacity of 

.

The cost per square foot of the material for the side of the barrel is half that of the cost per square foot for the top and bottom. 

Help him find the dimensions of the barrel that can be constructed at a minimum cost in terms of material used.