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)
$f'(75)$=$f(90)$-$f(60)$/$90-60$
$f'(75)$=($354.5-324.5$)/(90-60)=30/30=1 degF/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)$L(75)$=$f(75)+f'(75)(t-75)$
=$342.8+1(t-75)$

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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\)$F(72)$~$L(72)$
=$342.8+1(72-75)$
=$339.8$degF

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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)
I think the linear approximation is exactly right , because F(72) 339.8 is not to far from F(75)=342.8
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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)$F(100)~L(100)$
=$342.8+1(100-75)$
=$367.8 deg F$
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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)I think my estimate of F(100)=367.8 deg F is to to large because its to far from point F(75)=342.8

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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)

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