Solve using definite integrals The function p(x) = -0.3x^2 + 20 (x)^1/2 + 35 gives the price of a stock, in dollars, over
the first 25 days of the month.

b. Find the five-day moving average value of the stock through the 7th day (that
is, from x = 2 to x = 7 days). (Round your answer to the nearest cent.)

The five-day moving average of a stock price over the interval from day 2 to day 7 can be found by computing the average of the function $ p(x) $ over this interval using a definite integral. This is done as follows:

The formula for the average value of a continuous function $ p(x) $ over an interval $[a, b]$ is given by:

\[
\text{Average Value} = \frac{1}{b - a} \int_a^b p(x) \, dx
\]

Here, $ p(x) = -0.3x^2 + 20\sqrt{x} + 35 $, and we need to find the average value of $ p(x) $ from $ x = 2 $ to $ x = 7 $. So, $ a = 2 $ and $ b = 7 $.

### Step 1: Set up the integral

We compute the definite integral of $ p(x) $ from 2 to 7:

\[
\int_2^7 \left( -0.3x^2 + 20\sqrt{x} + 35 \right) dx
\]

### Step 2: Compute the average

To find the five-day moving average, we divide the result of the definite integral by $ b - a = 7 - 2 = 5 $.

Let's now compute the definite integral and average value.The five-day moving average value of the stock price from day 2 to day 7 is approximately **$70.14**.

Would you like more details or have any further questions?

Here are 5 related questions you might find useful:
1. How do we derive the formula for the average value of a function using integrals?
2. What does a five-day moving average represent in stock market analysis?
3. How do definite integrals relate to area under the curve in calculus?
4. Can the moving average formula be applied to non-continuous data?
5. How would the result change if the interval was from day 1 to day 6 instead?

**Tip:** When solving integrals involving square roots, it can help to recognize patterns in integration formulas like those for power functions!

Solve using definite integrals. The function p(x) = -0.3x^2 + 20(x)^1/2 + 35 gives the price of a stock, in dollars, over the first 25 days of the month. Find the five-day moving average value of the stock through the 7th day (that is, from x = 2 to x = 7 days). (Round your answer to the nearest cent.)

Learn how to calculate the five-day moving average of a stock price using definite integrals. This problem involves applying the average value of a function formula over an interval, with detailed steps for finding the integral of p(x) = -0.3x^2 + 20√x + 35 from day 2 to day 7. Suitable for advanced high school or college students.

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