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.
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(a)
| $t$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|--------|------|---|---|---|---|---|---|---|
| $P(t)$ | 1000 | 1100 |1210 | 1331 |1464 | 1610 | 1771 | 1948 |

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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)$P(t)=1002.29*1.09976^x+-2.26115$

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(c\) What will the population be after 100 years under this model?
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(c\)$P(100)=100e^{0.095\left(100\right)}$=13359727

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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 |
|--- |---|---|---|---|---|---|
| $P'(t)$ | 1105 |1221 | 1350 | 1492 | 1649 | 1822 |

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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) $P''(3)=(1002.29)(1.09976)^3ln(1.0999976)^2-2.26115=979606$

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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)

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

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(b) Find the proper dosage for a 128 lb individual.
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(b) $D(x)=0.025(128)^2+0.5(128)+10= 356$ dosages

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(c\) What is the interpretation of the value $D'(128)$.
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(c\) $D'(x)=0.05x-0.5$
$D'(128)=0.05x128-0.5$
$D'(128)=5.9$
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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) By using linear approximation formula
$y=f(a)+f'(a)(x-a)$
use the values from the table x=120 then the corresponding value of y=310 and a=128 since we need to find $D'(128)$
$y=D(a)+D'(a)(x-a)$
$310=D(128)+D'(128)(20-128)$
$310=356+D'(128)(-8)$
$D'(128)=(310-356)/(-8)=5.75$

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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) $y_{130}=D(130)=0.025(130)^2-0.5(130)+10$
$y_{130}=D(130)=367.5$
$y-367.5=6(x-130)$
$y-367.5=6x-780$
$6x-y=412.5$
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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) $6x-y=412.5$ at $x=128lbs$
$y=6x-412.5$
$y=6(128)-412.5$
$y=355.5$ or $y=356$
This is a good estimation

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