Chapter 8

Water is being poured into a bucket at a steady rate. \(h(t)\) gives the height of water at time \(t\). Let \(t_{*}\) be the time when the bucket is half full. What can you say about the signs of \(h^{\prime}\left(t_{*}\right)\) and \(h^{\prime \prime}\left(t_{*}\right) .\) Explain your reasoning precisely in plain English.

Use the tangent line approximation (linear approximation) of \(f(x)=\sqrt{x}\) at \(x=25\) to approximate the following. Use the graph of \(\sqrt{x}\) and its tangent line at \(x=25\) to predict the relative accuracy of your approximations. Check the accuracy using a computer or calculator. (a) \(\sqrt{23}\) (b) \(\sqrt{24}\) (c) \(\sqrt{24.9}\) (d) \(\sqrt{25.1}\) (e) \(\sqrt{26}\) (f) \(\sqrt{27}\)

For Problems 1 through 8 , find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=3 x^{2}+3 x+3+3 x^{-1}+3 x^{-2} $$

Suppose that the revenue, \(R\), brought in each month by the after-eight shows at a movie theater is a function of the price \(p\) of a ticket. Suppose that \(R\) is measured in thousands of dollars and that \(p\) is measured in dollars. Interpret the following statements in words. (a) \(R^{\prime}(3.5)=50\) (b) \(R^{\prime}(7.50)=-15\)

Suppose we want to use a tangent line approximation of \(f(x)=\sqrt{x}\) at \(x=a\) to approximate a particular square root numerically. Which values of \(a\) should we choose to approximate each of the following? (a) \(\sqrt{102}\) (b) \(\sqrt{8}\) (c) \(\sqrt{18}\) (d) \(\sqrt{115.5}\)

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{x-2 x^{2}}{5} $$

A company is making industrial-size rolls of paper towels. A machine is wrapping paper around a roll at a steady rate. By this we mean that the same number of sheets of paper towels are added to the roll every minute. Let \(D(t)\) be the diameter of the roll of paper towels at time \(t\). Determine the sign of the following. Explain your answers using plain English. (a) \(D(t)\) (b) \(D^{\prime}(t)\) (c) \(D^{\prime \prime}(t)\)

Use a tangent line approximation to \(f(x)=\frac{1}{x}\) at \(x=2\) to approximate \(\frac{1}{1.9}\).

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\pi\left(3 x^{2}+7 x+1\right)(x-2) $$

Let \(Y(t)\) be the number of Japanese yen exchangeable for one U.S. dollar, where \(t\) is the number of days after January 1,1996 . (a) What is the practical significance of the values of \(t\) for which \(Y^{\prime}(t)\) is positive? (b) What is the practical significance of the values of \(t\) for which \(Y^{\prime}(t)\) is negative and \(Y^{\prime \prime}(t)\) is negative? (c) What is the meaning of the statement \(Y^{\prime}(5)=0.8\) ? (d) Interpret the quantity \(\frac{Y(5)-Y(3)}{2}\).

Approximate \(\sqrt{98}\) using the appropriate rst derivative to help you. Explain your reasoning.

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{1}{x^{2}+4} $$

Use the fact that \(\frac{d}{d x} x^{1 / 3}=\left(\frac{1}{3}\right) x^{-2 / 3}\) to approximate \(\sqrt[3]{30}\). Do you expect your answer to be an over- approximation or an under-approximation? Explain. Compare your answer to the approximation supplied by your calculator.

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{x}{x+2} $$

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{x+2}{x} $$

Consider the solid right cylinder with a xed height of 10 inches and a variable radius. Let \(V(r)\) be the volume of the cylinder as a function of \(r\), the radius, given in inches. Interpret \(d V / d r\) geometrically. Explain why your answer makes sense by looking at \(\Delta V\) geometrically.

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\left(\frac{5 x^{2}}{2}+7 x^{5}-5 x\right) x $$

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\frac{a}{x}+b x(c-d x), \text { where } a, b, c, \text { and } d \text { are constants. } $$

Rewrite each of these functions in a form that allows you to differentiate using the tools you have now, but with as little exertion as possible. Then differentiate. Work on strategy; none of these problems require the Quotient Rule. (a) \(f(x)=\frac{\frac{1}{x}+1}{\frac{x+1}{x^{2}+2 x}}\) (b) \(f(x)=\frac{\left(x^{3}+3 x\right)}{2 x^{4}}\)

Rewrite each of these functions in a form that allows you to differentiate using the tools you have now, but with as little exertion as possible. Then differentiate. Work on strategy; none of these problems require the Quotient Rule. (a) \(f(x)=(x+1)(x-1) x\) (b) \(f(x)=\frac{x+\frac{1}{x}}{x}\)

For each function in Problems 12 through 14 : (a) Sketch the graph of \(f\). (b) Find \(f^{\prime}(x)\). Are there any values of \(x\) for which \(f^{\prime}\) is undefined? $$ f(x)=x+|x| $$

For each function : (a) Sketch the graph of \(f\). (b) Find \(f^{\prime}(x)\). Are there any values of \(x\) for which \(f^{\prime}\) is undefined? $$ f(x)=|x|-x $$

For each function : (a) Sketch the graph of \(f\). (b) Find \(f^{\prime}(x)\). Are there any values of \(x\) for which \(f^{\prime}\) is undefined? $$ f(x)=\left\\{\begin{array}{ll} x^{2}, & x \geq 0 \\ x, & x<0 \end{array}\right. $$

The function \(f(x)=x+|x|\) is continuous at \(x=0\) but not differentiable at \(x=0\). Explain, using the de nitions of continuity at a point and differentiability at a point.

Let \(f(x)=\frac{1}{x^{2}+1}\) (a) Sketch the graph of \(f .\) Do this by graphing \(x^{2}+1\) and looking at the reciprocal. Check your answer with a computer or calculator. (b) Make a rough sketch of \(f^{\prime}\) based on your graph of \(f\). (c) Find \(f^{\prime}(x)\) analytically, using the Quotient Rule. Graph \(f^{\prime}\).

Let \(f(x)=\left\\{\begin{array}{ll}x^{3}, & x \leq 1 \\ k x, & x>1\end{array}\right.\) (a) What values of \(k\) makes \(f(x)\) a continuous function? (b) If \(k\) is chosen so that \(f\) is continuous at \(x=1\), is \(f\) differentiable there?

Find the equation of the tangent line to \(f(x)=x\left(x^{2}+2\right)\) at \(x=1\).

For what value(s) of \(x\) is the slope of the tangent line to \(f(x)=\frac{1}{3} x^{3}\) equal to 1 ?

For Problems 20 through 23, find the following: $$ \frac{d}{d x}\left(\frac{x+1}{x^{3}+3 x+1}\right) $$

Find the following: $$ \frac{d}{d x}\left(\frac{\pi}{\pi x+\pi}\right) $$

Find the following: $$ \frac{d}{d x}\left(\frac{2 x^{2}+x+1}{\sqrt{2 x}}\right) $$

Find the following: $$ \frac{d}{d x}\left(\frac{x^{2}+5 x}{2 x^{10}}\right) $$