Chapter 2: Problem 9
Find \(d y / d x\) by implicit differentiation. \(x^{3}-3 x^{2} y+2 x y^{2}=12\)
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Area The radius \(r\) of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when (a) \(r=8\) centimeters and (b) \(r=32\) centimeters.
Adiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure \(p\) and volume \(V\) satisfy the equation \(p V^{1.3}=k,\) where \(k\) is a constant. Find the relationship between the related rates \(d p / d t\) and \(d V / d t .\)
Electricity The combined electrical resistance \(R\) of two resistors \(R_{1}\) and \(R_{2},\) connected in parallel, is given by $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ where \(R, R_{1},\) and \(R_{2}\) are measured in ohms. \(R_{1}\) and \(R_{2}\) are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate is \(R\) changing when \(R_{1}=50\) ohms and \(R_{2}=75\) ohms?
Modeling Data The table shows the health care expenditures \(h\) (in billions of dollars) in the United States and the population \(p\) (in millions) of the United States for the years 2004 through 2009 . The year is represented by \(t,\) with \(t=4\) corresponding to 2004 . (Source: U.S. Centers for Medicare \& Medicaid Services and U.S. Census Bureau) $$ \begin{array}{|c|c|c|c|c|c|}\hline \text { Year, } & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline h & {1773} & {1890} & {2017} & {2135} & {2234} & {2330} \\\ \hline p & {293} & {296} & {299} & {302} & {305} & {307} \\\ \hline\end{array} $$ (a) Use a graphing utility to find linear models for the health care expenditures \(h(t)\) and the population \(p(t) .\) (b) Use a graphing utility to graph each model found in part (a). (c) Find \(A=h(t) / p(t),\) then graph \(A\) using a graphing utility. What does this function represent? (d) Find and interpret \(A^{\prime}(t)\) in the context of these data.
Conjecture Consider the functions \(f(x)=x^{2}\) and \(g(x)=x^{3} .\) (a) Graph \(f\) and \(f^{\prime}\) on the same set of axes. (b) Graph \(g\) and \(g^{\prime}\) 'on the same set of axes. (c) Identify a pattern between \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h^{\prime}(x)\) if \(h(x)=x^{n},\) where \(n\) is an integer and \(n \geq 2\) . (d) Find \(f^{\prime}(x)\) if \(f(x)=x^{4}\) . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.
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