Function Analysis Part 2 Assignment.sagews

Function Analysis Part 2 Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Analyze the following functions using the steps from class.

Question 1

$f(x)=\displaystyle e^x\cdot\sqrt[3]{x^2+2x+1}$

[We'll work through this one together in class.]

Step 1: Find the domain of $f$. Discuss vertical asymptotes and holes.

Step 2: Find the derivative $f'$.

Step 3: Find the critical points of $f$ (where $f'$ is $0$ or undefined).

Step 4: See if the sign of $f'$ actually changes at the critical points of $f$, and determine whether $f$ has a local maximum or local minimum at these points.

Step 5: Find the second derivative $f''$.

Step 6: Find the critical points of $f'$ (where $f''$ is $0$ or undefined).

Step 7: See if the sign of $f''$ actually changes at the critical points of $f'$, and determine whether $f$ has an inflection point at these points.

Step 8: Find the $x$- and $y$-intercepts.

Step 9: Determine the end behavior.

Step 10: Make an informed graph. Mark any $x$- and $y$-intercepts, relative maxima and minima, and inflection points.

Step 11: Discuss absolute max/min, increasing/decreasing, concave up/down.

Question 2

$\displaystyle g(x)=\frac{6x^2-x-2}{2x^2+x-3}$

[Hint: One graph will not show all the important features.]

Step 1: Find the domain of $g$.

Step 2: Find the derivative $g'$.

Step 3: Find the critical points of $g$ (where $g'$ is $0$ or undefined).

Step 4: See if the sign of $g'$ actually changes at the critical points of $g$, and determine whether $g$ has a local maximum or local minimum at these points.

Step 5: Find the second derivative $g''$.

Step 6: Find the critical points of $g'$ (where $g''$ is $0$ or undefined).

Step 7: See if the sign of $g''$ actually changes at the critical points of $g'$, and determine whether $g$ has an inflection point at these points.

Step 8: Find the $x$- and $y$-intercepts.

Step 9: Determine the end behavior.

Step 10: Make an informed graph. Mark any $x$- and $y$-intercepts, relative maxima and minima, and inflection points.

Step 11: Discuss absolute max/min, increasing/decreasing, concave up/down.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.