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title: 'Math 181 Miniproject 4: Linear Approximation and Calculus'

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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) $AV[60,90]$
f(90)-f(60)/90-60
354.5-324.5/30
              =1 deg/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)$L(x) = 342.8+1(x-75)$


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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\) $L(72)=342.8+1(72-75)$
$L(72)=342.8+1(-3)$
L(72)=339.8 deg.F

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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)It is larger since F"(t) is decreasing.

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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)$L(100) = 342.8 + 1(100-75)$
$L(100)=342.8+1(25)$
$L(100)=367.8 deg.F$

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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 estimate the approximation to be larger, because we only have F(75) since we have the slope of it. Central differences for number lower can't be used, and we can't find it for F(90) since we have no other values.

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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) ![](https://i.imgur.com/GUkIlRj.png)
F(60) and F(90) are plotted on the line it would be a decent estimation of the values, but not for other points. The line approximation over estimates too much values under F(60).


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