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 | 1610.51 | 1771.561 | 1948.7171 |

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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) $f\left(x\right)=1000\left(1.10\right)^{x}$

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(c\) What will the population be after 100 years under this model?
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(c\) 13780612.34, rounding to 13780612 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) The value P'(5) is the rate of change or "average" of how many people are in the settlement per year after t, years. After 5 years, an additional year increases the population by 153.7, rounding to 154 people.
| $t$ | 1 | 2 | 3 | 4 | 5 | 6 |
|--- |---|---|---|---|---|---|
| $P'(t)$ | 105 |115.5 | 127.05 | 139.8 |153.7 | 169.1 |

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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) The interpretation of P"(3) is the slope at a point on the P'(3) graph. It is the acceleration found by being the derivative of the velocity. The rate of change of velocity with respect to time, t years. So after 3 years, an additional year increases the population by about 12 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) I think the value of k is 1000, meaning 1000 people.

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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) $D\left(x\right)=0.025x^{2}-0.5x+10$

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(b) Find the proper dosage for a 128 lb individual.
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(b) 355.6 mg.

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(c\) What is the interpretation of the value $D'(128)$.
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(c\) The interpretation of the value D'(128) is the rate of change or "average" of mg/lbs after x, lbs. With someone weighing 128 lbs, an additional pound would give them 5.9 mg of their dosage.

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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) I used the function, $D\left(x\right)=0.025x^{2}-0.5x+10$, and used the Central difference method to find the value of D'(x) at x being 128 pounds. So I plugged in 129 into the function and got my answer of 361.525. Then I plugged in 127 and got 349.725. I then used the Central difference formula and did (361.525-349.725)/(129-127)=5.9 lbs/mg.

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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) $y=367.5x+47781$

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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) Yes, it does give a good estimate for the dosage for a 128 lb individual.

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