Chapter 7

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\left\\{\begin{array}{ll}x^{2}-4, & x \leq 0 \\ 2 x+4, & x>0\end{array}\right.\)

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\sqrt{x+2} ;(7,3) $$

find the limit $$ \lim _{x \rightarrow 2} \frac{2-x}{x^{2}-4} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\sqrt{x^{2}-2 x+1} $$

find \(f^{\prime}(x)\). $$ f(x)=\frac{4 x^{3}-3 x^{2}+2 x+5}{x^{2}} $$

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{x} ;(1,1) $$

find the limit $$ \lim _{t \rightarrow 4} \frac{t+4}{t^{2}-16} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\left(4-3 x^{2}\right)^{-2 / 3} $$

The number \(N\) of gallons of regular unleaded gasoline sold by a gasoline station at a price of \(p\) dollars per gallon is given by \(N=f(p)\). (a) Describe the meaning of \(f^{\prime}(2.959)\) (b) Is \(f^{\prime}(2.959)\) usually positive or negative? Explain.

find \(f^{\prime}(x)\). $$ f(x)=\frac{-6 x^{3}+3 x^{2}-2 x+1}{x} $$

Find the constant \(a\) (Exercise 45\()\) and the constants \(a\) and \(b\) (Exercise 46 ) such that the function is continuous on the entire real line. \(f(x)=\left\\{\begin{array}{ll}2, & x \leq-1 \\ a x+b, & -1

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{x-1} ;(2,1) $$

find the limit $$ \lim _{t \rightarrow 1} \frac{t^{2}+t-2}{t^{2}-1} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\frac{\sqrt{x}+1}{x^{2}+1} $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x-1} $$

find \(f^{\prime}(x)\). $$ f(x)=x^{4 / 5}+x $$

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(h(x)=\frac{1}{x^{2}-x-2}\)

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ f(x)=-\frac{1}{4} x^{2} \quad x+y=0 $$

find the limit $$ \lim _{x \rightarrow-2} \frac{x^{3}+8}{x+2} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{2 x}{x+1}} $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

find \(f^{\prime}(x)\). $$ f(x)=x^{1 / 3}-1 $$

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(k(x)=\frac{x-4}{x^{2}-5 x+4}\)

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ f(x)=x^{2}+1 \quad 2 x+y=0 $$

find the limit $$ \lim _{x \rightarrow-1} \frac{x^{3}-1}{x+1} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{4}}{x^{3}+1} $$

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(f(x)=\left\\{\begin{array}{ll}2 x-4, & x \leq 3 \\ x^{2}-2 x, & x>3\end{array}\right.\)

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ f(x)=-\frac{1}{2} x^{3} \quad 6 x+y+4=0 $$

find the limit $$ \lim _{x \rightarrow-2} \frac{|x+2|}{x+2} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{x}\left(2-x^{2}\right) $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{4}+3}{x^{2}+1} $$

Find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ y=x^{3}+x \quad(-1,-2) $$

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(f(x)=\left\\{\begin{array}{ll}3 x-1, & x \leq 1 \\ x+1, & x>1\end{array}\right.\)

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ f(x)=x^{2}-x \quad x+2 y-6=0 $$

find the limit $$ \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{x-2} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x(x+1) $$

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=|x+3| $$

$$ \lim _{x \rightarrow 2} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} 4-x, & x \neq 2 \\ 0 & x=2 \end{array}\right. $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ s(t)=\frac{1}{t^{2}+3 t-1} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x^{2}(x+1) $$

Find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. $$ f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x \quad(-1,2) $$

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -value(s) at which the function is not continuous. Explain why the function is not continuous at the \(x\) -value(s). \(f(x)=\llbracket 2 x-1 \rrbracket\)

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\left|x^{2}-9\right| $$

find the limit $$ \lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} x^{2}+2, & x \neq 1 \\ 1, & x=1 \end{array}\right. $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=-\frac{4}{(t+2)^{2}} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2]\). $$ f(x)=x(x+1)(x-1) $$

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=-x^{4}+3 x^{2}-1 $$

Describe the interval(s) on which the function is continuous. \(f(x)=\frac{x}{x^{2}+1}\)