Problem 56
Evaluate \(\int \sin ^{2} x d x\).
Problem 57
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) \(\int\left(x^{2} \cos 2 x+x \sin 2 x\right) d x\)
Problem 58
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.
Problem 58
Let \(F_{0}(x)=x \sin x\) and \(F_{n+1}(x)=\int F_{n}(x) d x\). (a) Determine \(F_{1}(x), F_{2}(x), F_{3}(x)\), and \(F_{4}(x)\). (b) On the basis of part (a), conjecture the form of \(F_{16}(x)\).
Problem 59
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.
Problem 59
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) \(f(x)=\sin \left[\left(x^{2}-6 x+40\right) / 6\right]\)
Problem 60
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.
Problem 61
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?
Problem 62
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?
Problem 64
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.
