Chapter 4

Suppose that every customer order taken by the XYZ Company requires exacty 5 hours of labor for handling the paperwork; this length of time is fixed and does not vary from lot to lot. The total number of hours \(y\) required to manufacture and sell a lot of size \(x\) would then be \(y=(\) number of hours to produce a lot of size \(x)+5\) Some data on XYZ's bookcases are given in the following table. $$ \begin{array}{ccc} \hline \text { Order } & \text { Lot Size } x & \begin{array}{c} \text { Total Labor } \\ \text { Hours } y \end{array} \\ \hline 1 & 11 & 38 \\ 2 & 16 & 52 \\ 3 & 8 & 29 \\ 4 & 7 & 25 \\ 5 & 10 & 38 \\ \hline \end{array} $$ (a) From the description of the problem, the least-squares line should have 5 as its \(y\)-intercept. Find a formula for the value of the slope \(b\) that minimizes the sum of squares $$ S=\sum_{i=1}^{n}\left[y_{i}-\left(5+b x_{i}\right)\right]^{2} $$ (b) Use this formula to estimate the slope \(b\). (c) Use your least-squares line to predict the total number of labor hours to produce a lot consisting of 15 bookcases.

A car is stationary at a toll booth. Twenty minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Explain why the car must have exceeded 60 miles per hour at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Evaluate the indefinite integral $$ \int \sin ^{3}\left[\left(x^{2}+1\right)^{4}\right] \cos \left[\left(x^{2}+1\right)^{4}\right]\left(x^{2}+1\right)^{3} x d x $$

The fixed monthly cost of operating a plant that makes Zbars is \(\$ 7000\), while the cost of manufacturing each unit is \(\$ 100\). Write an expression for \(C(x)\), the total cost of making \(x\) Zbars in a month.

Show that if an object's position function is given by \(s(t)=a t^{2}+b t+c\), then the average velocity over the interval \([A, B]\) is equal to the instantaneous velocity at the midpoint of \([A, B]\).

The manufacturer of Zbars estimates that 100 units per month can be sold if the unit price is \(\$ 250\) and that sales will increase by 10 units for each \(\$ 5\) decrease in price. Write an expression for the price \(p(n)\) and the revenue \(R(n)\) if \(n\) units are sold in one month, \(n \geq 100\)

\text { Evaluate } \int \sin ^{2} x d x

\text { Evaluate } \int \sin ^{2} x d x

What are you able to deduce about the shape of a vase based on each of the following tables, which give measurements of the volume of the water as a function of the depth. \begin{tabular}{l|lcccccc|} \hline (a) Depth & 1 & 2 & 3 & 4 & 5 & 6 \\ Volume & 4 & 8 & 11 & 14 & 20 & 28 \\ \hline (b) Depth & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline Volume & 4 & 9 & 12 & 14 & 20 & 28 \\ \hline \end{tabular}

Some software packages can evaluate indefinite integrals. Use your software on each of the following. (a) \(\int 6 \sin (3(x-2)) d x\) (b) \(\int \sin ^{3}(x / 6) d x\) (c) \(\int\left(x^{2} \cos 2 x+x \sin 2 x\right) d x\)

The total cost of producing and selling \(x\) units of Xbars per month is \(C(x)=100+3.002 x-0.0001 x^{2}\). If the production level is 1600 units per month, find the average cost, \(C(x) / x\), of each unit and the marginal cost.

Use a graphing calculator or a CAS to plot the graph of each of the following functions on \([-1,7]\). Determine the coordinates of any global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=x \sqrt{x^{2}-6 x+40}\) (b) \(f(x)=\sqrt{|x|}\left(x^{2}-6 x+40\right)\) (c) \(f(x)=\sqrt{x^{2}-6 x+40} /(x-2)\) (d) \(f(x)=\sin \left[\left(x^{2}-6 x+40\right) / 6\right]\)

The total cost of producing and selling \(n\) units of a certain commodity per week is \(C(n)=1000+n^{2} / 1200\). Find the average cost, \(C(n) / n\), of each unit and the marginal cost at a production level of 800 units per week.

The total cost of producing and selling \(100 x\) units of a particular commodity per week is $$ C(x)=1000+33 x-9 x^{2}+x^{3} $$ Find (a) the level of production at which the marginal cost is a minimum, and (b) the minimum marginal cost.

A price function, \(p\), is defined by $$ p(x)=20+4 x-\frac{x^{2}}{3} $$ where \(x \geq 0\) is the number of units. (a) Find the total revenue function and the marginal revenue function. (b) On what interval is the total revenue increasing? (c) For what number \(x\) is the marginal revenue a maximum?

For the price function defined by $$ p(x)=(182-x / 36)^{1 / 2} $$ find the number of units \(x_{1}\) that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, \(x_{1}\), is sold?

For the price function given by $$ p(x)=800 /(x+3)-3 $$ find the number of units \(x_{1}\) that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, \(x_{1}\), is sold?

A riverboat company offers a fraternal organization a Fourth of July excursion with the understanding that there will be at least 400 passengers. The price of each ticket will be \(\$ 12.00\), and the company agrees to discount the price by \(\$ 0.20\) for each 10 passengers in excess of 400 . Write an expression for the price function \(p(x)\) and find the number \(x_{1}\) of passengers that makes the total revenue a maximum.

The XYZ Company manufactures wicker chairs. With its present machines, it has a maximum yearly output of 500 units. If it makes \(x\) chairs, it can set a price of \(p(x)=200-0.15 x\) dollars each and will have a total yearly cost of \(C(x)=5000+6 x-0.002 x^{2}\) dollars. The company has the opportunity to buy a new machine for \(\$ 4000\) with which the company can make up to an additional 250 chairs per year. The cost function for values of \(x\) between 500 and 750 is thus \(C(x)=9000+6 x-0.002 x^{2}\). Basing your analysis on the profit for the next year, answer the following questions. (a) Should the company purchase the additional machine? (b) What should be the level of production?

The ZEE Company makes zingos, which it markets at a price of \(p(x)=10-0.001 x\) dollars, where \(x\) is the number produced each month. Its total monthly cost is \(C(x)=200+4 x-0.01 x^{2}\). At peak production, it can make 300 units. What is its maximum monthly profit and what level of production gives this profit?

The arithmetic mean of the numbers \(a\) and \(b\) is \((a+b) / 2\), and the geometric mean of two positive numbers \(a\) and \(b\) is \(\sqrt{a b}\). Suppose that \(a>0\) and \(b>0\). (a) Show that \(\sqrt{a b} \leq(a+b) / 2\) holds by squaring both sides and simplifying. (b) Use calculus to show that \(\sqrt{a b} \leq(a+b) / 2\). Hint: Consider \(a\) to be fixed. Square both sides of the inequality and divide through by \(b\). Define the function \(F(b)=(a+b)^{2} / 4 b\). Show that \(F\) has its minimum at \(a\). (c) The geometric mean of three positive numbers \(a, b\), and \(c\) is \((a b c)^{1 / 3}\). Show that the analogous inequality holds: $$ (a b c)^{1 / 3} \leq \frac{a+b+c}{3} $$ Hint: Consider \(a\) and \(c\) to be fixed and define \(F(b)=\) \((a+b+c)^{3} / 27 b\). Show that \(F\) has a minimum at \(b=\) \((a+c) / 2\) and that this minimum is \([(a+c) / 2]^{2}\). Then use the result from (b).

Show that of all three-dimensional boxes with a given surface area, the cube has the greatest volume. Hint: The surface area is \(S=2(l w+l h+h w)\) and the volume is \(V=l w h\). Let \(a=l w, b=l h\), and \(c=h w\). Use the previous problem to show that \(\left(V^{2}\right)^{1 / 3} \leq S / 6\). When does equality hold?