91Ó°ÊÓ

Sales A company introduces a new product for which the number of units sold \(S\) is \(S(t)=200\left(5-\frac{9}{2+t}\right)\) where \(t\) is the time in months. (a) Find the average rate of change of \(S\) during the first year. (b) During what month of the first year does \(S^{\prime}(t)\) equal the average rate of change?

Converse of Rolle's Theorem Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b) .\) If there exists \(c\) in \((a, b)\) such that \(f^{\prime}(c)=0,\) does it follow that \(f(a)=f(b) ?\) Explain.

Rolle's Theorem Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b) .\) Also, suppose that \(f(a)=f(b)\) and that \(c\) is a real number in the interval \((a, b)\) such that \(f^{\prime}(c)=0 .\) Find an interval for the function \(g\) over which Rolle's Theorem can be applied, and find the corresponding critical number of \(g\) , where \(k\) is a constant. (a) $$g(x)=f(x)+k$$ (b) $$g(x)=f(x-k)$$ (c) $$g(x)=f(k x)$$

Temperature When an object is removed from a furnace and placed in an environment with a constant temperature of \(90^{\circ} \mathrm{F},\) its core temperature is \(1500^{\circ} \mathrm{F} .\) Five hours later, the core temperature is \(390^{\circ} \mathrm{F}\) . Explain why there must exist a time in the interval \((0,5)\) when the temperature is decreasing at a rate of \(222^{\circ} \mathrm{F}\) per hour.

Lawn Sprinkler A lawn sprinkler is constructed in such a way that \(d \theta / d t\) is constant, where \(\theta\) ranges between \(45^{\circ}\) and \(135^{\circ}(\) see figure). The distance the water travels horizontally is $$x=\frac{v^{2} \sin 2 \theta}{32}, \quad 45^{\circ} \leq \theta \leq 135^{\circ}$$ where \(v\) is the speed of the water. Find \(d x / d t\) and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water? FOR FURTHER INFORMATION For more information on the "calculus of lawn sprinklers," see the article "Design of an Oscillating Sprinkler" by Bart Braden in Mathematics Magazine. To view this article, go to MathArticles.com.

Highway Design In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are 9\(\%\) and 6\(\%\) (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points \(A\) and \(B\) . The horizontal distances from \(A\) to the \(y\) -axis and from \(B\) to the \(y\) -axis are both 500 feet. (a) Find the coordinates of \(A\) and \(B\) (b) Find a quadratic function \(y=a x^{2}+b x+c\) for \(-500 \leq x \leq 500\) that describes the top of the filled region. (c) Construct a table giving the depths \(d\) of the fill for \(x=-500,-400,-300,-200,-100,0,100,200,300\) \(400,\) and 500 . (d) What will be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together?

If \(x=c\) is a critical number of the function \(f,\) then it is also a critical number of the function \(g(x)=f(x)+k,\) where \(k\) is a a constant.

Increasing Functions Is the product of two increasing functions always increasing? Explain.

Critical Numbers Consider the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d,\) where \(a \neq 0 .\) Show that \(f\) can have zero, one, or two critical numbers and give an example of each case.

True or False? In Exercises \(75-78\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.\(\begin{array}{l}{\text { The graph of every cubic polynomial has precisely one point }} \\ {\text { of inflection. }}\end{array}\)