Math 181 Miniproject 3: Texting Lesson.md
---
My lesson Topic
===

<style>
body {
background-color: #eeeeee;
}

h1 {
color: maroon;
margin-left: 40px;
}

.gray {
margin-left: 50px ;
margin-right: 29%;
font-weight: 500;
color: #000000;
background-color: #cccccc;
border-color: #aaaaaa;
}

.blue {
display: inline-block;
margin-left: 29% ;
margin-right: 0%;
width: -webkit-calc(70% - 50px);
width: -moz-calc(70% - 50px);
width: calc(70% - 50px);
font-weight: 500;
color: #fff;
border-color: #336699;
background-color: #337799;
}

.left {
content:url("https://i.imgur.com/rUsxo7j.png");
width:50px;
border-radius: 50%;
float:left;
}
.right{
content:url("https://i.imgur.com/5ALcyl3.png"); width:50px;
border-radius: 50%;
display: inline-block;
vertical-align:top;
}
</style>

<div id="container" style=" padding: 6px;
color: #fff;
border-color: #336699;
background-color: #337799;
display: flex;
justify-content: space-between;
margin-bottom:3px;">
<div>
<i class="fa fa-envelope fa-2x"></i>
</div>
<div>
<i class="fa fa-camera fa-2x"></i>
</div>

<div>
<i class="fa fa-comments fa-2x"></i>
</div>
<div>
<i class="fa fa-address-card fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-phone fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-list-ul fa-2x" aria-hidden="true"></i>
</div>
<div>
<i class="fa fa-user-plus fa-2x" aria-hidden="true"></i>
</div>
</div>

<div><img class="left"/><div class="alert gray">

So... what exactly is this assignment? it looks tricky.

</div></div>

<div><div class="alert blue">

Choose one of the sample exam problem types and write a tutorial explaining what concept is being tested and how you might solve it. The tutorial will be in the form of a text message thread where you answer questions from an inquiring student.

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

I see. All this text inside the html `div` tags is where we add the text.

</div></div>

<div><img class="left"/><div class="alert gray">

What needs to be included?

</div></div>

<div><div class="alert blue">

To obtain a grade of **PASS** your work should include
* an explanation of the type of problem that is being tested and why it is important,
* a demonstration problem that is different than any example we have covered in class,
* a full solution of your demonstration problem, and
* any study tips or hints that will help a student to succeed on this problem on an exam.

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

How detailed do we need to be?

</div></div>

<div><div class="alert blue">

The inquiring student should make comments at the level of a member of the standard audience for Math 181. Your responses should be at the level of the instructor or a competent tutor. Be sure to follow the guidelines for earning a **PASS** at our [course home page](http://sergeballif.github.io/NSC-Math-181/Specifications.html).

</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

Can I get all fancy and use graphics and math formulas?

</div></div>

<div><div class="alert blue">

Yes. Just be sure to leave a blank line between your text and the `div` tags. Place math between dollar signs like $$f(x)=\sqrt{\frac{x+7}{x^2-24}}.$$ If you're feeling adventurous, you can try editing the html code at the top to get different colors or to change the profile pictures.

Also, don't forget to delete all the sample conversation.

</div><img class="right"/></div>

---

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

Math 181 Miniproject 2: Population and Dosage.md
---
Math 181 Miniproject 2: Population and Dosage
===

**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--1.6 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\. A settlement starts out with a population of 1000. Each year the population increases by $10\%$. Let $P(t)$ be the function that gives the population in the settlement after $t$ years.
(a) Find the missing values in the table below.
:::

(a)
| $t$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|--------|------|---|---|---|---|---|---|---|
| $P(t)$ | 1000 | 1100| 1210 | 1331 | 1464.1| 1610.51| 1771.561 | 1948.7171 |

:::info
(b) Find a formula for $P(t)$. You can reason it out directly or you can have Desmos find it for you by creating the table of values above (using $x_1$ and $y_1$ as the column labels) and noting that the exponential growth of the data should be modeled using an exponential model of the form
\\[
y_1\sim a\cdot b^{x_1}+c
\\]
:::

(b)
$P(t)\sim1000\cdot 1.1^{t}+1.2464\cdot10^{-12}$

:::info
(c\) What will the population be after 100 years under this model?
:::

(c\)
$P(100)\sim1000\cdot 1.1^{100}+1.2464\cdot10^{-12}$
$P(100)\sim13780612.3398$

:::info
(d) Use a central difference to estimate the values of $P'(t)$ in the table below. What is the interpretation of the value $P'(5)$?
:::

(d)
| $t$ | 1 | 2 | 3 | 4 | 5 | 6 |
|--- |---|---|---|---|---|---|
| $P'(t)$ | 105 | 115.5 | 127.05|139.755 | 153.7305| 169.10355 |

$P'(5)$ tells us that at 5 years the population is increasing 153.7305 persons per year.

:::info
(e) Use a central difference to estimate the values of $P''(3)$. What is the interpretation of this value?
:::

(e)
$P''(3)\sim\frac{P'(4)-P'(2)}{4-2}$
$P''(3)\sim\frac{139.755-115.5}{2}$
$P''(3)\sim12.1275$
This value shows the rate at which the exponential growth of the population is changing.

:::info
(f) **Cool Fact:** There is a constant $k$ such that $P'(t)=k\cdot P(t)$. In other words, $P$ and $P'$ are multiples of each other.

What is the value of $k$? (You could try creating a slider and playing with the graphs or you can try an algebraic approach.)
:::

(f)
$k=\frac{21}{220}$

:::success
2\. The dosage recommendations for a certain drug are based on weight.
| Weight (lbs)| 20 | 40 | 60 | 80 | 100 | 120 | 140 | 160 | 180 |
|--- |--- |--- |--- |--- |--- |--- |--- |--- |--- |
| Dosage (mg) | 10 | 30 | 70 | 130 | 210 | 310 | 430 | 570 | 730 |
(a) Find a function D(x) that approximates the dosage when you input the weight of the individual. (Make a table in Desmos using $x_1$ and $y_1$ as the column labels and you will see that the points seem to form a parabola. Use Desmos to find a model of the form
\\[
y_1\sim ax_1^2+bx_1+c
\\]
and define $D(x)=ax^2+bx+c$.)
:::

(a)
$D(x)=0.025x^2-0.5x+10$

:::success
(b) Find the proper dosage for a 128 lb individual.
:::

(b)
$D(x)=0.025\left(128\right)^2-0.5\left(128\right)+10$
$D(x)=355.6$ mg

:::success
(c\) What is the interpretation of the value $D'(128)$.
:::

(c\)
$D'(128)$ is the rate that the dosage changes at 128 lbs.

:::success
(d) Estimate the value of $D'(128)$ using viable techniques from our calculus class. Be sure to explain how you came up with your estimate.
:::

(d)
$D'(128)=\frac{D(140)-D(100)}{140-100}$
$D'(128)=\frac{430-210}{140-100}$
$D'(128)=5.5$

What I did is use a central difference of $P(140)$ and $P(100)$ to estimate the value of $P'(128)$. I used these values because they are given in the table.
:::success
(e) Given the value $D'(130)=6$, find an equation of the tangent line to the curve $y=D(x)$ at the point where $x=130$ lbs.
:::

(e)
$y(x)=D(130)+D'(130)(a-130)$
$y(x)=367.5+6(a-130)$
:::success
(f) Find the point on the tangent line in the previous part that has $x$-coordinate $x=128$. Does the output value on the tangent line for $x=128$ lbs give a good estimate for the dosage for a 128 lb individual?
:::

(f)
$y(128)=367.5+6(128-130)$
$y(128)=355.5$
This gives a good estimate that may be slightly lower than the true value.

---

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