91Ó°ÊÓ

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

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) \(\lim _{x \rightarrow 3} f(x)\).

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

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

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

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

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) \(\lim _{x \rightarrow-1} f(x)\).

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

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