Chapter 2

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt{x^{2}-3 x+1} $$

1-4. Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 2^{-}}(5 x-7)\) b. \(\lim _{x \rightarrow 2^{+}}(5 x-7)\) c. \(\lim _{x \rightarrow 2}(5 x-7)\)

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{4} \cdot x^{6} $$

Find the derivative of each function. $$ f(x)=x^{4} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \left(5 x^{2}-x+2\right)^{4} $$

Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 4^{-}}(2 x+1)\) b. \(\lim _{x \rightarrow 4^{+}}(2 x+1)\) c. \(\lim _{x \rightarrow 4}(2 x+1)\)

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{7} \cdot x^{2} $$

Find the derivative of each function. $$ f(x)=x^{5} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \left(x^{2}-x\right)^{-3} $$

Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 1^{-}}\left(\frac{x^{3}-1}{x-1}\right)\) b. \(\lim _{x \rightarrow 1^{+}}\left(\frac{x^{3}-1}{x-1}\right)\) c. \(\lim _{x \rightarrow 1}\left(\frac{x^{3}-1}{x-1}\right)\)

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{4}\left(x^{5}+1\right) $$

Find the derivative of each function. $$ f(x)=x^{500} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{1}{x^{2}+x} $$

Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 1^{-}}\left(\frac{x^{4}-1}{x-1}\right)\) b. \(\lim _{x \rightarrow 1^{-}}\left(\frac{x^{4}-1}{x-1}\right)\) c. \(\lim _{x \rightarrow 1}\left(\frac{x^{4}-1}{x-1}\right)\)

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{5}\left(x^{4}+1\right) $$

Find the derivative of each function. $$ f(x)=x^{1000} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{x^{3}+1}{x^{3}-1} $$

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=x^{2}\left(x^{3}+1\right) $$

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=|2 x|\)

Find the derivative of each function. $$ f(x)=x^{1 / 2} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{\sqrt{x}-1}{\sqrt{x}+1} $$

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=x^{3}\left(x^{2}+1\right) $$

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=|3 x|\)

Find the derivative of each function. $$ f(x)=x^{1 / 3} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \left(\frac{x+1}{x-1}\right)^{4} $$

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=x\left(5 x^{2}-1\right) $$

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=x^{2 / 5}\)

For each function, find: a. \(f^{\prime \prime}(x)\) and b. \(f^{\prime \prime}(3)\). $$ f(x)=\frac{x-1}{x} $$

Find the derivative of each function. $$ g(x)=\frac{1}{2} x^{4} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt{\frac{x-1}{x+1}} $$

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=x^{4 / 5}\)

For each function, find: a. \(f^{\prime \prime}(x)\) and b. \(f^{\prime \prime}(3)\). $$ f(x)=\frac{x+2}{x} $$

Find the derivative of each function. $$ f(x)=\frac{1}{3} x^{9} $$

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=2 x\left(x^{4}+1\right) $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt{x^{2}-9}+5 $$

For each function, find: a. \(f^{\prime \prime}(x)\) and b. \(f^{\prime \prime}(3)\). $$ f(x)=\frac{x+1}{2 x} $$

Find the average rate of change of the given function between the following pairs of \(x\) -values. [Hint: See pages 95-96.] a. \(x=1\) and \(x=3\) b. \(x=1\) and \(x=2\) c. \(x=1\) and \(x=1.5\) d. \(x=1\) and \(x=1.1\) e. \(x=1\) and \(x=1.01\) f. What number do your answers seem to be approaching? $$ \text {} f(x)=x^{2}+x $$

Find the derivative of each function. $$ g(w)=6 \sqrt[3]{w} $$

9-12. Find each limit by graphing the function and using TRACE or TABLE to examine the graph near the indicated \(x\) -value. \(\lim _{x \rightarrow 1} \frac{\frac{1}{x}-1}{1-x}\) Use window \([0,2]\) by \([0,5]\)

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=\sqrt{x}(6 x+2) $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \sqrt[3]{x^{3}+8}-5 $$

For each function, find: a. \(f^{\prime \prime}(x)\) and b. \(f^{\prime \prime}(3)\). $$ f(x)=\frac{x-2}{4 x} $$

Find the average rate of change of the given function between the following pairs of \(x\) -values. [Hint: See pages 95-96.] a. \(x=1\) and \(x=3\) b. \(x=1\) and \(x=2\) c. \(x=1\) and \(x=1.5\) d. \(x=1\) and \(x=1.1\) e. \(x=1\) and \(x=1.01\) f. What number do your answers seem to be approaching? $$ \underline{\phantom{xxx}} f(x)=2 x^{2}+5 $$

Find the derivative of each function. $$ g(w)=12 \sqrt{w} $$

Find each limit by graphing the function and using TRACE or TABLE to examine the graph near the indicated \(x\) -value. \(\lim _{x \rightarrow 1.5} \frac{2 x^{2}-4.5}{x-1.5}\) Use window \([0,3]\) by \([0,10]\).

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=6 \sqrt[3]{x}(2 x+1) $$

Use the Generalized Power Rule to find the derivative of each function. $$ f(x)=\left(x^{2}+1\right)^{3} $$

Find the average rate of change of the given function between the following pairs of \(x\) -values. [Hint: See pages 95-96.] a. \(x=2\) and \(x=4\) b. \(x=2\) and \(x=3\) c. \(x=2\) and \(x=2.5\) d. \(x=2\) and \(x=2.1\) e. \(x=2\) and \(x=2.01\) f. What number do your answers seem to be approaching? $$ \underline{\phantom{xxx}} f(x)=2 x^{2}+x-2 $$

For each function, find: a. \(f^{\prime \prime}(x)\) and b. \(f^{\prime \prime}(3)\). $$ f(x)=\frac{1}{6 x^{2}} $$

a. Show that the definition of the derivative applied to the function \(f(x)=\sqrt{x}\) at \(x=0\) gives \(f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{\sqrt{h}}{h}\). b. Use a calculator to evaluate the difference quotient \(\frac{\sqrt{h}}{h}\) for the following values of \(h\) : \(0.1,0.001\), and \(0.00001 .\) [Hint: Enter the calculation into your calculator with \(h\) replaced by \(0.1\), and then change the value of \(h\) by inserting zeros.] c. From your answers to part (b), does the limit exist? Does the derivative of \(f(x)=\sqrt{x}\) at \(x=0\) exist? d. Graph \(f(x)=\sqrt{x}\) on the window \([0,1]\) by \([0,1]\). Do you see why the slope at \(x=0\) does not exist?