Math 181 Miniproject 7: The Shape of a Graph.md
---
---
tags: MATH 181
---

Math 181 Miniproject 7: The Shape of a Graph
===

**Overview:** In this miniproject you will be using the techniques of calculus to find the behavior of a graph.

**Prerequisites:** The project draws heavily from the ideas of Chapter 1 and $2.8$ together with ideas and techniques of the first and second derivative tests from $3.1$.

---

:::info
We are given the functions
$$
f(x)=\frac{12x^2-16}{x^3},\qquad f'(x)=-\frac{12(x^2-4)}{x^4},\qquad f''(x)=\frac{24(x^2-8)}{x^5}.
$$
The questions below are about the function $f(x)$. Answer parts (1) through (10) below. If the requested feature is missing, then explain why. Be sure to include the work/test that you used to rigorously reach your conclusion. It is not sufficient to refer to the graph.
(1) State the function's domain.
:::

(1)
[-∞,0) u (0, ∞)

:::info
(2) Find all $x$- and $y$-intercepts.
:::

(2)
x-intercept: (12x^2 - 16)/x^3 = 0

y-intercept: 12(0)^2-16 / (x)^3 = -16/0 = 0

:::info
(3) Find all equations of horizontal asymptotes.
:::

(3) 12x^2 - 16 / x^3 = numerator is less than denominator ∴ Horizontal asymptote: y=0

:::info
(4) Find all equations of vertical asymptotes.
:::

(4) denominator = 0
(0)^3 = 0

:::info
(5) Find the interval(s) where $f$ is increasing.
:::

(5) increasing at [-∞, -2] and [2, ∞]
12(x^2 -4) / x^4 = 0
x = -2, 2, 0
[-∞, -2] = increasing at -2 switch from increasing to decreasing (local max)
[-2, 0] = decreasing
[0, 2] = decreasing at 2 switch from decreasing to increasing (local min)
[2, ∞] = increasing

:::info
(6) Find the $x$-value(s) of all local maxima. (Find exact values, and not decimal representations)
:::

(6) -2

:::info
(7) Find the $x$-value(s) of all local minima. (Find exact values, and not decimal representations)
:::

(7) 2

:::info
(8) Find the interval(s) on which the graph is concave downward.
:::

(8) 24(x^2 - 8)/ x^5 = 0
2√2 , 0

[-∞, -2√2] concave down
[-2√2, 0] concave up
[0, 2√2] concave down
[2√2, ∞] concave up

:::info
(9) State the $x$-value(s) of all inflection points. (Find exact values, and not decimal representations)
:::

(9) 2√2

:::info
(10) Include a sketch of the graph of $y=f(x)$. Plot the different segments of the graph using the color code below.
* **blue:** $f'>0$ and $f''>0$
* **red:** $f'<0$ and $f''>0$
* **black:** $f'>0$ and $f''<0$
* **gold:** $f'<0$ and $f''<0$

(In Desmos you could restrict the plot $y=f(x)$ on the interval $[2,3]$ by typing $y=f(x)\{2\le x\le 3\}$.) Be sure to set the bounds on the graph so that the features of the graph that you listed above are easy to see.
:::

(10) Purple line is original function.


---

To submit this assignment click on the Publish button . Then copy the url of the final document and submit it in Canvas.