Chapter 12: Chapter 12

For a rectangle with perimeter 20 to have the largest area, what dimensions should it have?

The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula \(q=100 e^{-3 p^{2}+p}\), where \(q\) is the demand in monthly sales and \(p\) is the retail price in yen. a. Determine the price elasticity of demand \(E\) when the retail price is set at \(¥ 3\) and interpret your answer. b. At what price will revenue be a maximum? c. Approximately how many paint-by-number sets will be sold per month at the price in part (b)?

Calculate \(\frac{d^{2} y}{d x^{2}}\). \(y=\frac{1}{x}-\ln x\)

The area of a circular sun spot is growing at a rate of $1,200 \mathrm{~km}^{2} / \mathrm{s}$ a. How fast is the radius growing at the instant when it equals $10,000 \mathrm{~km}$ ? HINT [See Example 1.] b. How fast is the radius growing at the instant when the sun spot has an area of \(640,000 \mathrm{~km}^{2} ?\) HINT [Use the area formula to determine the radius at that instant.]

Complete the following: If the graph of a function is concave up on its entire domain, then its second derivative is ________ on the domain.

Daily sales of Kent's Tents reached a maximum in January 2002 and declined to a minimum in January 2003 before starting to climb again. The graph of daily sales shows a point of inflection at June 2002 . What is the significance of the point of inflection?

Company A's profits satisfy \(P(0)=\$ 1\) million, \(P^{\prime}(0)=\) \$1 million per year, and \(P^{\prime \prime}(0)=-\$ 1\) million per year per year. Company B's profits satisfy \(P(0)=\$ 1\) million, \(P^{\prime}(0)=\) \(-\$ 1\) million per year, and \(P^{\prime \prime}(0)=\$ 1\) million per year per year. There are no points of inflection in either company's profit curve. Sketch two pairs of profit curves: one in which Company A ultimately outperforms Company B and another in which Company B ultimately outperforms Company A.

Company C's profits satisfy \(P(0)=\$ 1\) million, \(P^{\prime}(0)=\) \(\$ 1\) million per year, and \(P^{\prime \prime}(0)=-\$ 1\) million per year per year. Company D's profits satisfy \(P(0)=\$ 0\) million, \(P^{\prime}(0)=\) \(\$ 0\) million per year, and \(P^{\prime \prime}(0)=\$ 1\) million per year per year. There are no points of inflection in either company's profit curve. Sketch two pairs of profit curves: one in which Company C ultimately outperforms Company \(\mathrm{D}\) and another in which Company D ultimately outperforms Company C.

Explain geometrically why the derivative of a function has a relative extremum at a point of inflection, if it is defined there. Which points of inflection give rise to relative maxima in the derivative?

If we regard position, \(s\), as a function of time, \(t\), what is the significance of the third derivative, \(s^{\prime \prime \prime}(t) ?\) Describe an everyday scenario in which this arises.