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# System prepended metadata

title: 'Math 181 Miniproject 2: Population and Dosage'

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Math 181 Miniproject 2: Population and Dosage.md
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Math 181 Miniproject 2: Population and Dosage
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**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. 

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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|1610.5|1771.6|1948.7|





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(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
\\]
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(b)
$P(t)\sim1000\cdot1.1^{t}+\left(1.2757\ \cdot10^{-12}\right)$



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(c\) What will the population be after 100 years under this model?
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(c\)
$P(t)=13,780,612.3398$ people



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(d) Use a central difference to estimate the values of $P'(t)$ in the table below. What is the interpretation of the value $P'(5)$?
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(d)
| $t$     | 1 | 2   | 3    | 4    | 5    | 6   |
|---      |---|-----|------|------|------|-----|
| $P'(t)$ |105|115.5|127.05|139.75|153.75|169.1|
After 5 years, the population will increase approx. 153.75 people per year.




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(e) Use a central difference to estimate the values of $P''(3)$. What is the interpretation of this value?
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(e)
$P''\left(3\right)=\frac{P'\left(2\right)-P'\left(4\right)}{2-4}$
$=\frac{115.5-139.75}{-2}$
$=12.125\frac{people}{year}$
At three years, the rate of change in population is accelerating approx. 12.125 people per year.

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(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.)
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(f)
$P'(t)=k\cdot P(t)$
$105=k\cdot 1100$
$0.0954545=k$



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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 $D(x)=ax^2+bx+c$.)
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(a)
$y_1\sim0.025x_1^2+-0.5x_1+10$
$D(x)=0.025x^2+-0.5x+10$



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(b) Find the proper dosage for a 128 lb individual.
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(b)
$D(128)=0.025(128)^2+-0.5(128)+10$
$D(x)=355.6$mg




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(c\) What is the interpretation of the value $D'(128)$. 
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(c\)
$D'(128)$ represents the dosage in miligrams that will be added for every pound soomeone gains after 128lbs.

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(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.
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(d)
Using the limit definition, I will estimate the derivative function at 128.
$D'\left(128\right)=\frac{D\left(128+h\right)-D\left(128\right)}{h}$
$D'(128)=5.9\frac{mg}{lbs}$



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(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. 
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(e)
$y=D\left(130\right)+D'\left(130\right)\left(x-130\right)$
$y=367.5+6\left(x-130\right)$


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(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? 
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(f)
$y=367.5+6\left(x-130\right)$
$y=367.5+6\left(128-130\right)$
$y=355.5$
This value gives a good estimate for the dosage of a 128lb individual.



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