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)
\\[
y=1002.29\cdot 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\)The population after 100 years would be 13514042.8208 $\frac{people}{year}$.

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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 |
|--- |---|---|---|---|---|---|
| $P'(t)$ | 95.3| 115.2 | 126.7| 139.42 | 153.3 | 168.6 |

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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) Using the central difference estimate the values of P''(3)= 12.1. This can be interpreted that this is the rate of which the increase of population of the 3rd $\frac{people}{\frac{year}{year}}$.

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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)$P'(t)=k\cdot P(t)$
$P'(1)=k\cdot P(1)$
$95.3=k\cdot 1100$
$\frac{95.3}{1100}$ = k
k = 0.086

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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)The proper dosage is 355.6 mg for a 128 lb individual.

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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 5.9 $\frac{mb}{lbs}$, the weight per unit of 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) y = f(a)+f'(a)(x-a)
(120,310) a=128

y = f(a)+f'(a)(x-a)
310 = D(128) + D'(128)(120-128)
310 = 356 + D'(128)- 8
D'(128)= $\frac{310-356}{-8}$
= 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
$\left(\left(y_{130}\right)\right)=m \left(x-x_0\right)$
y - 367.5 = 6(x-130)
The equation of the tangent line is 367.5 + 6(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) 367.5 + 6(x-130) at x = 128 lbs
y = 367.5 + 6(x-130)
y = 367.5 + 6(128-130)
y = 367.5 + 6(-2)
y = 367.5 + -12
y = 355.5
The estimation dosage for 128 lbs is 356 which is a good estimation.

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