Function Analysis Part 1 Assignment.sagews

Function Analysis Part 1 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

$\displaystyle f(x)=\frac{9x-1}{\frac{1}{5}x^2+22}$

[We'll work through this one together in class. Look out for the three inflection points - they may not be obvious on our first graph.]

Step 1: Find the domain of $f$.

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

$g(x)=2x^4-8x^3-x^2+30x$

[Caution: $g$ has two x-intercepts. When you solve $g(x)=0$, Sage will give you four answers, but only two are real. Convert to decimals and watch out for scientific notation.]

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.