Chapter 1

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). $$f(x)=2 x+3$$

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. \(y=(3 \sqrt{x}+2) x^{2}\)

Classify each statement as either true or false. If \(\lim _{x \rightarrow 4} F(x)\) exists, then \(F\) must be continuous at \(x=4\)

Complete each of the following statements. The notation__________ is read “the limit, as x approaches 2 from the right.”

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

Find \(\frac{d y}{d x}\). $$ y=2 x^{15} $$

Find \(d^{2} y / d x^{2}\) $$ y=6 x-3 $$

Differentiate each function $$ y=\sqrt{4 x^{2}+1} $$

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). $$f(x)=-2 x+5$$

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. \(y=(4 \sqrt{x}+3) x^{3}\)

Classify each statement as either true or false. If \(\lim _{x \rightarrow 7} G(x)\) equals \(G(7),\) then \(G\) must be continuous at \(x=7\)

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=2 x+3 $$

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

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

Differentiate each function $$ y=\left(4 x^{2}+1\right)^{-50} $$

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). $$f(x)=\frac{3}{4} x-2$$

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. \(f(x)=(2 x+5)\left(3 x^{2}-4 x+1\right)\)

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2}(4 x-5) $$

Complete each of the following statements. The notation ______ is read “the limit as x approaches 5.”

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

Find \(\frac{d y}{d x}\). $$ y=x^{-6} $$

Find \(d^{2} y / d x^{2}\) $$ y=\frac{1}{x^{2}} $$

Differentiate each function $$ y=\left(8 x^{2}-6\right)^{-40} $$

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). $$f(x)=\frac{1}{2} x-3$$

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. \(g(x)=(4 x-3)\left(2 x^{2}+3 x+5\right)\)

Complete each of the following statements. The notation______ is read “the limit as x approaches \(\frac{1}{2}\)."

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 1}(3 x+2) $$

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=12 x^{3} $$

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

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

Differentiate each function $$ y=(x-4)^{8}(2 x+3)^{6} $$

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). $$f(x)=x^{2}+x$$

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. \(F(t)=(\sqrt{t}+2)(3 t-4 \sqrt{t}+7)\)

Consider the function \(f\) given \(b y\) $$ f(x)=\left\\{\begin{array}{ll} x-2, & \text { for } x \leq 3, \\ x-1, & \text { for } x>3. \end{array}\right. $$ If a limit does not exist, state that fact. Find (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x) ;\) (c) \(\lim _{x \rightarrow 3} f(x)\).

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-1}\left(x^{2}-4\right) $$

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=1-x^{3} $$

Find \(\frac{d y}{d x}\). $$ y=4 x^{-2} $$

Find \(d^{2} y / d x^{2}\) $$ y=\sqrt[4]{x} $$

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). $$f(x)=x^{2}-x$$

Differentiate each function $$ y=(x+5)^{7}(4 x-1)^{10} $$

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. \(G(t)=(2 t+3 \sqrt{t}+5)(\sqrt{t}+4)\)

Consider the function \(f\) given \(b y\) $$ f(x)=\left\\{\begin{array}{ll} x-2, & \text { for } x \leq 3, \\ x-1, & \text { for } x>3. \end{array}\right. $$ If a limit does not exist, state that fact. Find (a) \(\lim _{x \rightarrow-1^{-}} f(x) ;\) (b) \(\lim _{x \rightarrow-1^{+}} f(x) ;\) (c) \(\lim _{x \rightarrow-1} f(x)\).

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-2}\left(x^{2}+3\right) $$

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

Find \(\frac{d y}{d x}\). $$ y=x^{4}-7 x $$

Find \(f^{\prime \prime}(x)\) $$ f(x)=x^{3}-\frac{5}{x} $$

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). $$f(x)=-5 x^{2}-2 x+7$$

Differentiate each function $$ y=\frac{1}{(4 x+5)^{2}} $$

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(y=\frac{x^{6}}{x^{4}}\)

Consider the function g given by $$ g(x)=\left\\{\begin{array}{ll} x+6, & \text { for } x<-2, \\ -\frac{1}{2} x+1, & \text { for } x \geq-2. \end{array}\right. $$ If a limit does not exist, state that fact. Find (a) \(\lim _{x \rightarrow-2^{-}} g(x) ;\) (b) \(\lim _{x \rightarrow-2^{+}} g(x) ;\) (c) \(\lim _{x \rightarrow-2} g(x)\).