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### Question 4: The following table gives the temperature of an oven as it heats up.

| Time (min)       | 0  | 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12  | 13  | 14  |
|------------------|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| Temperature (°F) | 70 | 125 | 170 | 210 | 250 | 280 | 310 | 335 | 360 | 380 | 400 | 415 | 430 | 440 | 445 |

#### (a) **Graph the data.**
We need to plot time (min) on the x-axis and temperature (°F) on the y-axis.

#### (b) **Draw a curve of best fit and the tangent line at \(x = 5\).**
- The curve of best fit would involve finding a smooth curve that best fits the temperature data points.
- The tangent line at \(x = 5\) (time = 5 minutes, where temperature = 280°F) should have the same slope as the curve at that point.

#### (c) **Determine the slope of the tangent line using the \(y\)-intercept of the tangent line and the point of tangency (5, 280).**
To find the slope of the tangent line at \(x = 5\), we need to use the formula for the derivative or the rate of change at that point. The slope can also be calculated using the data around \(x = 5\).

#### (d) **Estimate the instantaneous rate of change in temperature at exactly 5 min using a centered interval from the table of values.**
The centered difference approximation for the rate of change is given by:
\[
\frac{f(x+h) - f(x-h)}{2h}
\]
We can approximate the derivative at \(x = 5\) using values from the table, such as using \(x = 4\) and \(x = 6\).

#### (e) **Compare your answers to parts (c) and (d).**
The values obtained from parts (c) and (d) should be similar if calculated correctly.

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Do you want me to help with plotting the graph or calculating the tangent slope and instantaneous rate of change for parts (b), (c), or (d)? 

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Here are 5 related questions you might find interesting:
1. How do you determine the equation of a tangent line for non-linear curves?
2. What is the significance of the rate of change in real-life applications of temperature?
3. How can we model oven temperature rise using a polynomial or exponential function?
4. How would errors in the measurements affect the curve of best fit?
5. What other methods could be used to approximate the derivative at a specific point?

**Tip**: The centered difference method is often used to approximate derivatives when exact formulas are unavailable or difficult to compute!

The following table gives the temperature of an oven as it heats up.

Learn how to graph oven temperature data, draw curves of best fit, calculate tangent lines, and estimate the instantaneous rate of change using centered interval approximations. Includes practical application of derivatives and mean value theorem.

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