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| 1464.1 |1610.5 | 1771.55| 1948.705|

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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) $P(t)=1000.04*1.1^x-0.0413883$

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(c\) What will the population be after 100 years under this model?
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(c\) $P(t)=1000.04*1.1^(100)+0.0413883$
1000.04*13780.61234+0.0413883
= 13,781,163 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)
$P'(t)=ln(1.1)*1000.04*1.1^x$
| $t$ | 1 | 2 | 3 | 4 | 5 | 6 |
|--- |---|---|---|---|---|---|
| $P'(t)$ | 104.854 | 115.329 | 126.863 | 139.549 | 153.504 | 168.855 |

$P'(5)$ is interpreted as the rate of change per year for the 5th year, over 150 new members of the population.

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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) units of P'(t)/units of x
$P"(3)=(153.504-126.863)/(4-5)
(26.641)/2
= 13.3205
This means in year three the rate of the population increase was incrested by about 13 people.

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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) K would be the beginning number of people, so 1000.

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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)$D(x)=x^2+x+1$
When inserting the form into desmos, it said the values of a, b, and c were 1.
The definition of D(x)=ax^2 =bx+c is the simplified version of the same formula because the input values were 1 for a b and c.

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(b) Find the proper dosage for a 128 lb individual.
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(b)
I plugged 128 into my formula and the output value was 16513. I know this is incorrect becaue by looking at the chart of values I know the value should be in between 10 and 30 mg.

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(c\) What is the interpretation of the value $D'(128)$.
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(c\) The interprettion of that value would be how many more milligrams of medicine to give the 128 pound person.

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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)
Im going to use the vales I already have with 120 pounds getting a 310 mg dose and 140 lbs getting 430 mg. I will find the central difference.
(430-310)/20
=6

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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)I looked at the value on desomos. I think the output at x=128 is fair.

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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)

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