Chapter 3: Problem 32
In Exercises \(29-32,\) find \(y^{\prime \prime}\) $$y=9 \tan (x / 3)$$
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True or False The derivative of \(y=\sqrt[3]{x}\) is \(\frac{1}{3 x^{2 / 3}} .\) Justify your answer.
Marginal Revenue Suppose the weekly revenue in dollars from selling x custom- made office desks is \(r(x)=2000\left(1-\frac{1}{x+1}\right)\) (a) Draw the graph of \(r .\) What values of \(x\) make sense in this problem situation? (b) Find the marginal revenue when \(x\) desks are sold. (c) Use the function \(r^{\prime}(x)\) to estimate the increase in revenue that will result from increasing sales from 5 desks a week to 6 desks a week. (d) Writing to Learn Find the limit of \(r^{\prime}(x)\) as \(x \rightarrow \infty\) How would you interpret this number?
Radians vs. Degrees What happens to the derivatives of \(\sin x\) and cos \(x\) if \(x\) is measured in degrees instead of radians? To find out, take the following steps. (a) With your grapher in degree mode, graph \(f(h)=\frac{\sin h}{h}\) and estimate \(\lim _{h \rightarrow 0} f(h) .\) Compare your estimate with \(\pi / 180 .\) Is there any reason to believe the limit should be \(\pi / 180 ?\) (b) With your grapher in degree mode, estimate \(\lim _{h \rightarrow 0} \frac{\cos h-1}{h}\) (c) Now go back to the derivation of the formula for the derivative of sin \(x\) in the text and carry out the steps of the derivation using degree-mode limits. What formula do you obtain for the derivative? (d) Derive the formula for the derivative of cos \(x\) using degree-mode limits. (e) The disadvantages of the degree-mode formulas become apparent as you start taking derivatives of higher order. What are the second and third degree-mode derivatives of \(\sin x\) and \(\cos x\) ?
Finding Tangents (a) Show that the tangent to the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ at the point \(\left(x_{1}, y_{1}\right)\) has equation $$\frac{x_{1} x}{a^{2}}+\frac{y_{1} y}{b^{2}}=1$$ (b) Find an equation for the tangent to the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ at the point \(\left(x_{1}, y_{1}\right)\)
Exploration Let \(y_{1}=a^{x}, y_{2}=\mathrm{NDER} y_{1}, y_{3}=y_{2} / y_{1},\) and \(y_{4}=e^{y_{3}}\) (a) Describe the graph of \(y_{4}\) for \(a=2,3,4,5 .\) Generalize your description to an arbitrary \(a>1\) (b) Describe the graph of \(y_{3}\) for \(a=2,3,4,\) 5. Compare a table of values for \(y_{3}\) for \(a=2,3,4,5\) with \(\ln a\) . Generalize your description to an arbitrary \(a>1\) (c) Explain how parts (a) and (b) support the statement \(\frac{d}{d x} a^{x}=a^{x} \quad\) if and only if \(\quad a=e\) (d) Show algebraically that \(y_{1}=y_{2}\) if and only if \(a=e\) .
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