Applications of the Derivative

Determine the graph of a function f from the graph of its derivative f'.

Determine the graph of a function f from the graph of its derivative f'. 

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Find the possibility graph of its derivative f'.

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{9}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{x}\mathbf{-}\mathbf{5}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{ }\frac{\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{2}\mathbf{)}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{x}}}{\mathbf{1}\mathbf{+}{\mathbf{e}}^{\mathbf{x}}}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{l}\mathit{n}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{l}\mathit{n}\left(ln\left(x\right)\right)$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\left(\frac{\pi }{2}x\right)$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{.}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathbf{(}\mathit{x}\mathbf{-}\mathbf{2}\mathbf{)}}^{\mathbf{2}}\mathbf{(}\mathbf{1}\mathbf{+}\mathit{x}\mathbf{)}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathit{x}\mathbf{+}\mathbf{1}\mathbf{)}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{x}\mathbf{-}\mathbf{2}}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}}{\mathbf{x}\mathbf{+}\mathbf{2}}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{x}}}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{\pi }\mathbf{\right(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\left)}\mathbf{\right)}$

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{\pi }\mathit{x}\mathbf{\right)}$

Use the first-derivative test to determine the local extrema of each function $f$ in Exercises $39-50$. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)=arc\tan x$.

Use the first-derivative test to determine the local extrema of each function $f$ in Exercises $39-50$. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)={\sin }^{-1}{x}^2$.

Use the first-derivative test to determine the local extrema of each function $f$ in Exercises $39-50$. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)=\sin \left({\cos }^{-1}x\right)$.

Use the first-derivative test to determine the local extrema of each function $f$ in Exercises $39-50$. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f\left(x\right)=\cos \left({\sin }^{-1}x\right)$.

For each sign chart for $f^{\text{'}}$ in Exercises $51-56$, sketch possible graphs of both $f^{\text{'}}$  and $f$ . On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$ is zero and $x$-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

For each sign chart for $f^{\text{'}}$ in Exercises 51–56, sketch possible graphs of both $f^{\text{'}}$and $f$ . On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$ is zero and $x$-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

For each sign chart for $f^{\text{'}}$ in Exercises $51-56$, sketch possible graphs of both $f^{\text{'}}$ and $f$. On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$ is zero and $x$-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

For each sign chart for $f^{\text{'}}$ in Exercises $51-56$, sketch possible graphs of both $f^{\text{'}}$ and $f$ . On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$ is zero and $x$-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

For each sign chart for $f^{\text{'}}$ in Exercises $51-56$, sketch possible graphs of both $f^{\text{'}}$ and $f$. On each sign chart, unlabeled tick-marks are locations where $f^{\text{'}}\left(x\right)$ is zero and $x$-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

For each sign chart for $f^{\text{'}}$ in Exercises $51-56$, sketch possible graphs of both $f^{\text{'}}$ and $f$ . On each sign chart, unlabeled tick-marks are locations where f(x) is zero and x-values where $f^{\text{'}}\left(x\right)$ does not exist are indicated by tick-marks labeled “DNE.”

Sketch careful, labeled graphs of each function f in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right) = (x - 2)(3x + 1)$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x^2-x+100$.

Sketch careful, labeled graphs of each function f in Exercises 57–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x^3-x^2-x+1$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x^3-9x+1$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x^3-6x^2+12x$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=x\left(x^2-4\right)$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of f .

$f\left(x\right)=(2x + 11)(x^2+10)$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points

and behaviors of $f$.

$f\left(x\right)=3x^5-10x^4$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)={\left(1-x^4\right)}^7$.

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of f and $f^{\text{'}}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\sqrt{x}-\frac{1}{\sqrt{x}}$.