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Chapter 7: Problem 56
Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\frac{x^{2}}{x^{2}-4} $$
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Credit Card Rate The average annual rate \(r\) (in percent form) for commercial bank credit cards from 2000 through 2005 can be modeled by \(r=\sqrt{-1.7409 t^{4}+18.070 t^{3}-52.68 t^{2}+10.9 t+249}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. (a) Find the derivative of this model. Which differentiation rule(s) did you use? (b) Use a graphing utility to graph the derivative on the interval \(0 \leq t \leq 5\). (c) Use the trace feature to find the years during which the finance rate was changing the most. (d) Use the trace feature to find the years during which the finance rate was changing the least.
The temperature \(T\) (in degrees Fahrenheit) of food placed in a refrigerator is modeled by \(T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right)\) where \(t\) is the time (in hours). What is the initial temperature of the food? Find the rates of change of \(T\) with respect to \(t\) when (a) \(t=1\), (b) \(t=3\), (c) \(t=5\), and (d) \(t=10\).
Use the demand function to find the rate of change in the demand \(x\) for the given price \(p\). $$ x=300-p-\frac{2 p}{p+1}, p=\$ 3 $$
Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=g(x)+h(x) $$
The ordering and transportation cost \(C\) per unit (in thousands of dollars) of the components used in manufacturing a product is given by $$ C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad 1 \leq x $$ where \(x\) is the order size (in hundreds). Find the rate of change of \(C\) with respect to \(x\) for each order size. What do these rates of change imply about increasing the size of an order? Of the given order sizes, which would you choose? Explain. (a) \(x=10\) (b) \(x=15\) (c) \(x=20\)
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