Chapter 1

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=5 x^{2} $$

Find \(d^{2} y / d x^{2}\) $$ y=x^{4}-7 $$

Find \(\frac{d y}{d x}\) $$ y=x^{8} $$

Differentiate each function $$ y=(3-2 x)^{2} $$ Check by expanding and then differentiating.

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=\frac{1}{2} x^{2}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(y=x^{9} \cdot x^{4}\)

Complete each of the following statements. As x approaches______________ , the value of -3x approaches 6.

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=4 x^{2} $$

Find \(d^{2} y / d x^{2}\) $$ y=x^{5}+9 $$

Find \(\frac{d y}{d x}\). $$ y=x^{7} $$

Differentiate each function $$ y=(2 x+1)^{2} $$ Check by expanding and then differentiating.

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=\frac{3}{2} x^{2}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(y=x^{5} \cdot x^{6}\)

Classify each statement as either true or false. $$ \text { If } \lim _{x \rightarrow 2} f(x)=9, \text { then } \lim _{x \rightarrow 2} \sqrt{f(x)}=3 $$

Complete each of the following statements. As x approaches_________ , the value of x - 2 approaches 5.

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=-5 x^{2} $$

Find \(d^{2} y / d x^{2}\) $$ y=2 x^{4}-5 x $$

Find \(\frac{d y}{d x}\). $$ y=-0.5 x $$

Differentiate each function $$ y=(7-x)^{55} $$

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=-2 x^{2}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(f(x)=(2 x+5)(3 x-4)\)

Classify each statement as either true or false. If \(\lim _{x \rightarrow 1} g(x)=5,\) then \(\lim _{x \rightarrow 1}[g(x)]^{2}=25\)

Complete each of the following statements. The notation \(\lim _{x \rightarrow 4} f(x)\) is read________.

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=-4 x^{2} $$

Find \(\frac{d y}{d x}\). $$ y=-3 x $$

Find \(d^{2} y / d x^{2}\) $$ y=5 x^{3}+4 x $$

Differentiate each function $$ y=(8-x)^{100} $$

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=-3 x^{2}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(g(x)=(3 x-2)(4 x+1)\)

Classify each statement as either true or false. If \(\lim _{x \rightarrow 4} F(x)=7,\) then \(\lim _{x \rightarrow 4}[c \cdot F(x)]=7 c\)

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=x^{2}-x $$

Find \(\frac{d y}{d x}\). $$ y=7 $$

Find \(d^{2} y / d x^{2}\) $$ y=4 x^{2}-5 x+7 $$

Differentiate each function $$ y=\sqrt{1-x} $$

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=-x^{3}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(F(x)=3 x^{4}\left(x^{2}-4 x\right)\)

Classify each statement as either true or false. If \(g\) is discontinuous at \(x=3,\) then \(g(3)\) must not exist.

Complete each of the following statements. The notation \(\lim _{x \rightarrow 5^{-}} F(x)\) is read______.

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=x^{2}+x $$

Find \(\frac{d y}{d x}\). $$ y=12 $$

Find \(d^{2} y / d x^{2}\) $$ y=4 x^{2}+3 x-1 $$

Differentiate each function $$ y=\sqrt{1+8 x} $$

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=x^{3}$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(G(x)=4 x^{2}\left(x^{3}+5 x\right)\)

Classify each statement as either true or false. If \(f\) is continuous at \(x=2,\) then \(f(2)\) must exist.

Complete each of the following statements. The notation \(\lim _{x \rightarrow 4^{+}} G(x)\) is read___________.

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=\frac{9}{x} $$

Find \(\frac{d y}{d x}\) $$ y=3 x^{10} $$

Find \(d^{2} y / d x^{2}\) $$ y=7 x+2 $$

Differentiate each function $$ y=\sqrt{3 x^{2}-4} $$