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The problem you're dealing with shows the number of iTunes songs purchased (in millions) at different times (in days) after Apple iTunes sold its first million songs. Let's break down the questions and solve them step by step.

### Part A: Estimate \( S'(1396) \)
This question asks you to estimate the rate of change of the number of songs purchased when \( t = 1396 \) days. This can be approximated by using the difference quotient formula for the derivative:

\[
S'(t) \approx \frac{S(t + \Delta t) - S(t)}{\Delta t}
\]

For \( t = 1396 \), use nearby points from the table to estimate the derivative. The closest points are:
- \( t = 1077 \) with \( S(1077) = 1300 \) million songs
- \( t = 1396 \) with \( S(1396) = 2000 \) million songs

We will use these two points to estimate the slope:

\[
S'(1396) \approx \frac{S(1396) - S(1077)}{1396 - 1077} = \frac{2000 - 1300}{1396 - 1077} = \frac{700}{319} \approx 2.19 \, \text{million songs/day}
\]

So, the estimated rate of change \( S'(1396) \) is approximately **2.19 million songs/day**.

### Part B: Estimate \( S(1600) \)
This part asks for an estimate of the number of songs purchased when \( t = 1600 \) days. Since 1600 days is not in the table, you can use interpolation or extrapolation. One simple approach is linear extrapolation based on the values at \( t = 1396 \) and \( t = 1077 \).

Using the slope calculated earlier for \( t = 1396 \), we can extrapolate to estimate \( S(1600) \). The difference between 1600 and 1396 is:

\[
1600 - 1396 = 204 \, \text{days}
\]

Assuming the rate of change remains roughly the same:

\[
S(1600) \approx S(1396) + S'(1396) \times 204 = 2000 + 2.19 \times 204 \approx 2000 + 446.76 = 2446.76 \, \text{million songs}
\]

Thus, \( S(1600) \) is approximately **2446.76 million songs**.

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Would you like more details on these steps or any clarifications?

### Relative Questions:
1. How would the estimate of \( S'(t) \) change if you used a smaller time interval for the derivative?
2. What other interpolation methods could be used for estimating \( S(1600) \)?
3. How would quadratic interpolation differ from linear extrapolation in estimating \( S(1600) \)?
4. Why is it important to consider the accuracy of rate of change estimates in real-world data?
5. How could you use the second derivative to improve the accuracy of your estimates for \( S(1600) \)?

### Tip:
When estimating derivatives from tables, using smaller intervals between points can often provide a more accurate result.

In 2007, Apple's iTunes music store sold 2 billion songs. The number of iTunes songs purchased (in millions) is shown on the following chart, S(t), where time is measured in days since Apple iTunes sold 1 million songs (March 15, 2003). Estimate S'(1396) and S(1600).

Learn how to estimate the rate of change (S'(t)) of iTunes song purchases using derivative approximation and extrapolate values for S(1600). This problem involves calculus concepts and real-world data interpretation, suitable for high school or college students.

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