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Chapter 7: Problem 5
Find the derivative of the function. $$ y=3 $$
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The value \(V\) of a machine \(t\) years after it is purchased is inversely proportional to the square root of \(t+1\). The initial value of the machine is (a) Write \(V\) as a function of \(t\). (b) Find the rate of depreciation when \(t=1\). (c) Find the rate of depreciation when \(t=3\).
Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$
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 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) \(x=20\)
Use the General Power Rule to find the derivative of the function. $$ f(x)=-3 \sqrt[4]{2-9 x} $$
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