Math 181 Miniproject 4: Linear Approximation and Calculus.md
---
Math 181 Miniproject 4: Linear Approximation and Calculus
===

**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**.

---

:::info
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.
:::

(a)$$AV [60,90]$$

$$=
\frac{f\left(60\right)-f\left(90\right)}{(90-60)}$$

$$=
\frac{f\left(354.5\right)-f\left(324.5\right)}{(30)}$$
$$=1°F/minute$$

:::info
(b) Find the local linearization $y = L(t)$ to the function $y = F(t)$ at the point where $a = 75$.
:::

(b) $$L(x)=342.8+1(x-75)$$

:::info
(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.
:::

(c\)
$$ L(72)=342.8+1(72-75$$
$$L(72)=342.8+1(-3)$$
$$L(72)=339.8°F$$
:::info
(d) Do you think your estimate in (c) is too large, too small, or exactly right? Why?
:::

(d) f"(t) is decreasing so the estimate in (c) is too large.

:::info
(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.
:::

(e)$$L(100)=342.8+1(100-75)$$
$$L(100)=342.8+1(25)$$
$$L(100)=367.8°F$$

:::info
(f) Do you think your estimate in (e) is too large, too small, or exactly right? Why?
:::

(f) The estimate in (c) is exactly right because at t=75 line of local linearization touches the curve.

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

https://www.desmos.com/calculator/zrouq8wcop
(g) f(60) and f(90) are plotted on the line so this is a good approximation at or near t=75 but for the other values L(t) is and overestimate.

---

To submit this assignment click on the Publish button . Then copy the url of the final document and submit it in Canvas.