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)$$P(t)=P_0(1+r)^t$$
The Growth Rate Formula would give $$P_0$$ equals the initial amount, t as time interval, and r the rate of change.

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(c\) What will the population be after 100 years under this model?
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(c\)$$P(t)=P_0(1+r)^t$$
$$P(100)=1000(1+.10)^100$$
$$P(100)=1000(1.10)^100$$
$$P(100)=13,780,612.34$$

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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|139.755|153.731|169.1036|

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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)$$P''(3)=(139.775-115.5)/(4-2)$$
$$P''(3)=(24.255)/(2)$$
$$P''(3)=12.128 people/year^2$$

The rate will increase by 12.128 people/year^2 after 3 years

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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)The value would be 10% because it correlates to both functions P'(t) and P''(t). This can be seen when you plug in 10 to p'(t) which gives you 10.5. The same thing with P''(t) onlt that you get 115.5 as your answer.

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

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(b) Find the proper dosage for a 128 lb individual.
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(b)$$D(128)=0.025(128)^2-0.5(128)+10$$
$$D(128)=409.6-64+10$$
$$D(128)=355.6mg$$

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(c\) What is the interpretation of the value $D'(128)$.
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(c\) We get the number 355.6 mg at the point x=128 using the function: D(x)=0.025x^2-0.5x+10

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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)$$D'(x)=(P(b)-P(a))/(b-a)$$
$$D'(130)=(140)-120))/(140-120)$$
$$D'(130)=(430-310/20)$$
$$D'(130)=(120/20)$$
$$D'(130)=6mg/lb$$

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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)$$L(x)=d(a)+d'(a)(x-a)$$
$$L(x)=d(130)+d'(130)(x-130)$$
$$L(x)=[0.025(130)-0.5'(130)+10]+6(x-130)$$
$$L(x)=367.5+(x-130)$$

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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)$$L(x)=367.5+(x-130)$$
$$L(x)=367.5+(128-130)$$
$$L(x)=355.5mg$$

The output value we got (355.5mg) from this function is basically the same one as the origional dosage of 128 lb which was 355.6mg, making this a good estimate.
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