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

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Miniproject 4Math 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(75+15)-f(75-15)/2*15=f(90)-f(60)/30= 354.5-329.5/30=1.
F'(75)=1.

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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) y-f(a)=f'(a)(t-a)
y-342.8=1(t-75)
y=342.8+(t-75)
y=t+267.8
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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)
y=72+267.8=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) In the table, F(t)' is decreasing, thus F(t) is concave down, F''(t) should not be positive. This means that F(72)is too large.    

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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(t)=t+267.8
F(100)=100+267.8=367.8 degrees 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 that the temperature is appropriate, it's not too low or too high. 

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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) The line L(t) is a good approximation of F(t) at t(75)


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