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Problem 10

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=-6\)

Expert verified

The slope of the tangent line to the graph of the given function \(f(x) = -6\) at any point is 0, as the function is a constant and its derivative \(f'(x)\) is 0.

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Chapter 9

The number of Americans aged 45 to 54 is approximately $$ \begin{aligned} N(t)=&-0.00233 t^{4}+0.00633 t^{3}-0.05417 t^{2} \\ &+1.3467 t+25 \end{aligned} $$ million people in year \(t\), with \(t=0\) corresponding to the beginning of 1990 . Compute \(N^{\prime}(10)\) and \(N^{\prime \prime}(10)\) and interpret your results. Source U.S. Census Bureau

Chapter 9

Find the derivative of each function. \(f(x)=\left(3 x^{2}+2 x+1\right)^{-2}\)

Chapter 9

SocIALLY RESPONSIBLE FuNDS Since its inception in 1971 , socially responsible investments, or SRIs, have yielded returns to investors on par with investments in general. The assets of socially responsible funds (in billions of dollars) from 1991 through 2001 is given by $$ f(t)=23.7(0.2 t+1)^{1.32} \quad(0 \leq t \leq 11) $$ where \(t=0\) corresponds to the beginning of 1991 . a. Find the rate at which the assets of SRIs were changing at the beginning of 2000 . b. What were the assets of SRIs at the beginning of 2000 ?

Chapter 9

EFFECT OF HousING STARTS ON JoBS The president of a major housing construction firm claims that the number of construction jobs created is given by $$ N(x)=1.42 x $$ where \(x\) denotes the number of housing starts. Suppose the number of housing starts in the next \(t\) mo is expected to be $$ x(t)=\frac{7 t^{2}+140 t+700}{3 t^{2}+80 t+550} $$ million units/year. Find an expression that gives the rate at which the number of construction jobs will be created \(t\) mo from now. At what rate will construction jobs be created 1 yr from now?

Chapter 9

SELLING PRICE OF DVD RECORDERS The rise of digital music and the improvement to the DVD format are some of the reasons why the average selling price of standalone DVD recorders will drop in the coming years. The function $$ A(t)=\frac{699}{(t+1)^{0.94}} \quad(0 \leq t \leq 5) $$ gives the projected average selling price (in dollars) of stand-alone DVD recorders in year \(t\), where \(t=0\) corresponds to the beginning of 2002 . How fast was the average selling price of standalone DVD recorders falling at the beginning of \(2002 ?\) How fast was it falling at the beginning of \(2006 ?\)

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