Math 181 Miniproject 4: Linear Approximation and Calculus.md
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Math 181 Miniproject 4: Linear Approximation and Calculus
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**Overview:** In this miniproject you will put the idea of the *local linearization* of a function to build linear approximations to complex functions and then make *interpolations* and *extrapolations* using them.

**Prerequisites:** Sections 1.8 in *Active Calculus*, which focuses on this topic. **Completion of Miniprojects 1 and 2 is recommended before doing this miniproject**.

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1\. A potato is placed in an oven, and the potato's temperature $F$ (in degrees Fahrenheit) at various points in time is taken and recorded in the following table. The time $t$ is measured in minutes.
| $t$ | 0 | 15 | 30 | 45 | 60 | 75 | 90 |
|----- |---- |------- |----- |----- |------- |------- |------- |
| $F$ | 70 | 180.5 | 251 | 296 | 324.5 | 342.8 | 354.5 |

(a) Use a central difference to estimate $F'(75)$. Use this estimate as needed in subsequent questions in this problem.
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(a)This is the central difference formula we will use:
$$\frac{f(b)-
f(a)}{b-a}$$
We will use 90 and 60 to find the central difference.
$$f'(75)=\frac{354.5 - 324.5}{90-60}$$
$$f'(75)=\frac{
30}{30}$$
Our answer comes out to be$$1\frac{℉}{min}.$$

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(b) Find the local linearization $y = L(t)$ to the function $y = F(t)$ at the point where $a = 75$.
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(b)$$L(x)=f(a)+f'(a)(x-a)$$
$$L(x)=f(75)+f'(75)(x-75)$$
$$L(x)=342.8+1(x-75)$$

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(c\) Determine an estimate for $F(72)$ by employing the local linearization. Terminology: This estimate is called an *interpolation* because we are estimating a value that lies within a data set, between two known data points.
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(c\)$$L(x)=f(a)+f'(a)(x-a)$$
$$L(72)=f(75)+f'(75)(x-75)$$
$$L(72)=342.8+1(72-75)$$
$$L(72)=342.8-3$$
$$=339.8 ℉$$

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(d) Do you think your estimate in (c) is too large, too small, or exactly right? Why?
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(d)This estimate is small. It it a close guesstimate.

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(e) Use your local linearization to estimate $F(100)$. Terminology: This estimate is called an *extrapolation* because we are estimating a value that lies outside the range of values of a data set.
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(e)
$$L(72)=f(75)+f'(75)(x-75)$$
$$L(72)=342.8+1(100-75)$$
$$L(72)=342.8+25$$
$$=367.8 ℉$$

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(f) Do you think your estimate in (e) is too large, too small, or exactly right? Why?
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(f)The estimate is exactly right, the tangent line goes through $f(100)$ and $f(75)$.

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(g) Plot both $F$ and $L$ and comment on how or when the line $L(t)$ is a good approximation of $F(t)$.
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(g)
This is a good approximation of F(t).

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