Chapter 3

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{9}{x} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=x+\frac{1}{x}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=x+\frac{1}{x} $$

For the following exercises, use the definition of a derivative to find \(f^{\prime}(x)\) . $$f(x)=\frac{1}{\sqrt{x}}$$

Use the definition of a derivative to find \(f^{\prime}(x)\). $$ f(x)=\frac{1}{\sqrt{x}} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{(1+h)^{2 / 3}-1}{h}$$

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{(1+h)^{2 / 3}-1}{h} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{\left[3(2+h)^{2}+2\right]-14}{h}$$

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{\left[3(2+h)^{2}+2\right]-14}{h} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{(2+h)^{4}-16}{h}$$

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{(2+h)^{4}-16}{h} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{\left[2(3+h)^{2}-(3+h)\right]-15}{h}$$

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{\left[2(3+h)^{2}-(3+h)\right]-15}{h} $$

For the following exercises, the given limit represents the derivative of a function \(y=f(x)\) at \(x=a .\) Find \(f(x)\) and \(a .\) $$\lim _{h \rightarrow 0} \frac{e^{h}-1}{h}$$

For the following functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{l}{2 \sqrt{x}, 0 \leq x \leq 1} \\ {3 x-1, x>1}\end{array}\right.$$

For the functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1\). $$ f(x)=\left\\{\begin{array}{l} 2 \sqrt{x}, 0 \leq x \leq 1 \\ 3 x-1, x>1 \end{array}\right. $$

For the following functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{l}{3, x<1} \\ {3 x, x \geq 1}\end{array}\right.$$

For the functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1\). $$ f(x)=\left\\{\begin{array}{l} 3, x<1 \\ 3 x, x \geq 1 \end{array}\right. $$

For the following functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{l}{-x^{2}+2, x \leq 1} \\ {x, x>1}\end{array}\right.$$

For the functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1\). $$ f(x)=\left\\{\begin{array}{l} -x^{2}+2, x \leq 1 \\ x, x>1 \end{array}\right. $$

For the following functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1.\) $$f(x)=\left\\{\begin{array}{l}{2 x, x \leq 1} \\ {\frac{2}{x}, x>1}\end{array}\right.$$

For the functions, a. sketch the graph and b. use the definition of a derivative to show that the function is not differentiable at \(x=1\). $$ f(x)=\left\\{\begin{array}{l} 2 x, x \leq 1 \\ \frac{2}{x}, x>1 \end{array}\right. $$

For the following functions, use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x).\) $$f(x)=2-3 x$$

Use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x)\). $$ f(x)=2-3 x $$

For the following functions, use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x).\) $$f(x)=4 x^{2}$$

Use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x)\). $$ f(x)=4 x^{2} $$

For the following functions, use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x).\) $$f(x)=x+\frac{1}{x}$$

Use \(f^{\prime \prime}(x)=\lim _{h \rightarrow 0} \frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}\) to find \(f^{\prime \prime}(x)\). $$ f(x)=x+\frac{1}{x} $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=-\frac{5}{x}$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=-\frac{5}{x} $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=3 x^{2}+2 x+4$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=3 x^{2}+2 x+4 $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=\sqrt{x}+3 x$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=\sqrt{x}+3 x $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=\frac{1}{\sqrt{2 x}}$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=\frac{1}{\sqrt{2 x}} $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=1+x+\frac{1}{x}$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=1+x+\frac{1}{x} $$

For the following exercises, use a calculator to graph \(f(x)\) . Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$f(x)=x^{3}+1$$

Use a calculator to graph \(f(x)\). Determine the function \(f^{\prime}(x),\) then use a calculator to graph \(f^{\prime}(x)\). $$ f(x)=x^{3}+1 $$

Describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(P(x)\) denotes the population of a city at time \(x\) in years.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(\quad f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(C(x)\) denotes the total amount of money (in thousands of dollars) spent on concessions by \(x\) customers at an amusement park.

Describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(C(x)\) denotes the total amount of money (in thousands of dollars) spent on concessions by \(x\) customers at an amusement park.

Describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(R(x)\) denotes the total cost (in thousands of dollars) of manufacturing \(x\) clock radios.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(g(x)\) denotes the grade (in percentage points) received on a test, given \(x\) hours of studying.

Describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(g(x)\) denotes the grade (in percentage points) received on a test, given \(x\) hours of studying.

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(B(x)\) denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in \(x\) years since 1990 .

For the following exercises, describe what the two expressions represent in terms of each of the given situations. Be sure to include units. a. \(\frac{f(x+h)-f(x)}{h}\) b. \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(p(x)\) denotes atmospheric pressure at an altitude of \(x\) feet.