Chapter 7

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(6 x-5)^{4} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=x\left(x^{2}+3\right) $$

The table shows the amounts \(A\) (in billions of dollars per year) spent on \(\mathrm{R\&D}\) in the United States from 1980 through 2004 , where \(t\) is the year, with \(t=0\) corresponding to 1980 . Approximate the average rate of change of \(A\) during each period. (a) \(1980-1985\) (b) \(1985-1990\) (c) \(1990-1995\) (d) \(1995-2000\) (e) \(1980-2004\) (f) \(1990-2004\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline A & 63 & 72 & 81 & 90 & 102 & 115 & 120 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline A & 126 & 134 & 142 & 152 & 161 & 165 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline A & 166 & 169 & 184 & 197 & 212 & 228 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline A & 245 & 267 & 277 & 276 & 292 & 312 \\ \hline \end{array} $$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1)\). (a) \(y=x^{2}\) (b) \(y=x^{1 / 2}\)

Describe the interval(s) on which the function is continuous. Explain why the function is continuous on the interval(s). If the function has a discontinuity, identify the conditions of continuity that are not satisfied. \(f(x)=\frac{x^{3}-8}{x-2}\)

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=5 x^{3}-x^{2}+2\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2}(2 x+5) $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}-2 x+3\right)^{3} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ g(x)=(x-4)(x+2) $$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1)\). (a) \(y=x^{3 / 2}\) (b) \(y=x^{3}\)

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\left(x^{2}-1\right)^{3}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2}\left(x^{2}-3 x+1\right) $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(4-x^{2}\right)^{-1} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=x^{2}\left(3 x^{3}-1\right) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(t)=3 t+5 ;[1,2] $$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1)\). (a) \(y=x^{-1}\) (b) \(y=x^{-1 / 3}\)

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{1}{x^{2}-4}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\left(x^{2}+1\right)^{4 / 3} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\left(x^{2}+1\right)(2 x+5) \quad(-1,6) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ h(x)=2-x ;[0,2] $$

Find the slope of the tangent line to \(y=x^{n}\) at the point \((1,1)\). (a) \(y=x^{-1 / 2}\) (b) \(y=x^{-2}\)

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{1}{9-x^{2}}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-3 x+2} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\sqrt{5 x-2} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{1}{3}\left(2 x^{3}-4\right) \quad\left(0,-\frac{4}{3}\right) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ h(x)=x^{2}-4 x+2 ;[-2,2] $$

Find the derivative of the function. $$ y=3 $$

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{1}{4+x^{2}}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\sqrt{1-x^{2}} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{1}{7}\left(5-6 x^{2}\right) \quad\left(1,-\frac{1}{7}\right) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=x^{2}-6 x-1 ;[-1,3] $$

Find the derivative of the function. $$ f(x)=-2 $$

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{3 x}{x^{2}+1}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+2}-\sqrt{2}}{x} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(3 x+1)^{-1} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ g(x)=\left(x^{2}-4 x+3\right)(x-2) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=3 x^{4 / 3} ;[1,8] $$

Find the derivative of the function. $$ y=x^{4} $$

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0^{-}} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline x & -0.5 & -0.1 & -0.01 & -0.001 & 0 \\ \hline f(x) & & & & & ? \\ \hline \end{array} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=(x+2)^{-1 / 2} $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ g(x)=\left(x^{2}-2 x+1\right)\left(x^{3}-1\right) $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=x^{3 / 2} ;[1,4] $$

Find the derivative of the function. $$ h(x)=2 x^{5} $$

Determine whether the function is continuous on the entire real line. Explain your reasoning. \(f(x)=\frac{x+4}{x^{2}-6 x+5}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0^{-}} \frac{\frac{1}{2+x}-\frac{1}{2}}{2 x} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.5 & 0.1 & 0.01 & 0.001 & 0 \\ \hline f(x) & & & & & ? \\ \hline \end{array} $$

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{2}, u=4 x+7 $$

Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ h(x)=\frac{x}{x-5} $$

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\frac{1}{x} ;[1,4] $$