Math 181 Miniproject 1: Modeling and Calculus.md
---
Math 181 Miniproject 1: Modeling and Calculus
===

**Overview:** In this miniproject you will use technological tools to turn data and into models of real-world quantitative phenomena, then apply the principles of the derivative to them to extract information about how the quantitative relationship changes.

**Prerequisites:** Sections 1.1--1.5 in *Active Calculus*, specifically the concept of the derivative and how to construct estimates of the derivative using forward, backward and central differences. Also basic knowledge of how to use Desmos.

---

:::info
1\. The table below gives the distance that a car will travel after applying the brakes at a given speed.
| Speed (in mi/h) | Distance to stop (in ft) |
|----------------- |-------------------------- |
| 10 | 5 |
| 20 | 19 |
| 30 | 43 |
| 40 | 76.5 |
| 50 | 120 |
| 60 | 172 |
| 70 | 234 |

(a) Find a function $f(x)$ that outputs stopping distance when you input speed. This will just be an approximation. To obtain this function we will first make a table in Desmos. The columns should be labled $x_1$ and $y_1$. Note that the points are plotted nicely when you enter them into the table. Click on the wrench to change the scale of the graph to fit the data better. Since the graph has the shape of a parabola we hope to find a quadratric formula for $f(x)$. In a new cell in Desmos type
\\[
y_1\sim ax_1^2+bx_1+c
\\]
and let it come up with the best possible quadratic model. Use the suggested values of $a$, $b$, and $c$ to make a formula for $f(x)$.
:::

(a) The equation is$f$($x$)$=(0.048$x^2$)+(0.012$x$)+(-0.071).$

:::info
(b) Estimate the stopping distance for a car that is traveling 43 mi/h.
:::

(b) It will take the car 89.2 feet to come to a complete stop if it was traveling at 43 mi/h.
$f$($x$)$=0.048(43)^2+0.012(43)+(-0.071)$
$=89.20 ft$
:::info
(c\) Estimate the stopping distance for a car that is traveling 100 mi/h.
:::

(c ) If the car is traveling at 100 miles per hour, it will take 481.12 feet for the car to stop.
$f$($x$)$=0.048(100)$^2$+0.012(100)+(-0.071)$
$=481.12 ft$
:::info
(d) Use the interval $[40,50]$ and a central difference to estimate the value of $f'(45)$. What is the interpretation of this value?
:::

(d) If the car goes 45 miles per hour, the rate which it will result in the car stopping will increase by 4.35 feet for every mile per hour that the car goes faster.
$f'$($45$)=$AV(40,50)$=$\frac{f(50)-f(40)}{50-40}$= $\frac{120-76.5}{10}$= $4.35$ $\frac{ft}{mi/hr}$

:::info
(e) Use your function $f(x)$ on the interval $[44,46]$ and a central difference to estimate the value of $f'(45)$. How did this value compare to your estimate in the previous part?
:::

(e) The value when using the interval (44,46) has a difference of 0.06 which is very small difference.
$f'(45)$=$AV(44,46)$= $\frac{f(46)-f(44)}{46-44}$= $4.29$ $\frac{ft}{mi/hr}$
:::info
(f) Find the exact value of $f'(45)$ using the limit definition of derivative.
:::

$f'(x)=0.096x+0.012)$
$f'(45)=0.096(45)+0.012$
$f'(45)=4.33$ $\frac{ft}{mi/h}$

:::success
2\. Suppose that we want to know the number of squares inside a $50\times50$ grid. It doesn't seem practical to try to count them all. Notice that the squares come in many sizes.

(a) Let $g(x)$ be the function that gives the number of squares in an $x\times x$ grid. Then $g(3)=14$ because there are $9+4+1=14$ squares in a $3\times 3$ grid as pictured below.
Find $g(1)$, $g(2)$, $g(4)$, and $g(5)$.
:::

(a)
$g(1)=1$
$g(2)=5$
$g(4)=24$
$g(5)=55$
:::success
(b) Enter the input and output values of $g(x)$ into a table in Desmos. Then adjust the window to display the plotted data. Include an image of the plot of the data (which be exported from Desmos using the share button ). Be sure to label your axes appropriately using the settings under the wrench icon .
:::

(b) The data was input into desmos.

:::success
(c\) Use a cubic function to approximate the data by entering
\\[
y_1\sim ax_1^3+bx_1^2+cx_1+d
\\]
into a new cell of Desmos (assuming the columns are labeled $x_1$ and $y_1$). Find an exact formula for $g(x)$.
:::

(c\) The formula for g(x) is $g(x)=1.33x^3-8.07x^2+20.60x-13.2$

:::success
(d) How many squares are in a $50\times50$ grid?
:::

(d) There are 147,091.8 squares in a 50 x 50 grid.
$g(x)=1.33(50)^3-8.07(50)^2+20.60(50)-13.2$
$=147,091.8$

:::success
(e) How many squares are in a $2000\times2000$ grid?
:::

(e) There are $1.06$x$10^{10}$ squares in a 2000 x 2000 grid.
$g(x)=1.33(2000)^3-8.07(2000)^2+20.60(2000)-13.2$
=$1.06$×$10^{10}$

:::success
(f) Use a central difference on an appropriate interval to estimate $g'(4)$. What is the interpretation of this value?
:::

(f) Using the central difference, the outcome of $g'(4)$ is $21.21$.
$g'(4)=AV(3,5)$= $\frac{g(5)-g(3)}{5-3}$ $=21.21$


---

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