Given the function defined on the interval
below, find the value of where
the average value.

on the interval

Round to the nearest thousandth.

Ask by Deleon Boone. in the United States

Mar 22,2025

Question

Given the function defined on the interval
below, find the value of where
the average value.

on the interval

Round to the nearest thousandth.

Ask by Deleon Boone. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We start by finding the average value $$ f_{\text{avg}} $$ of the function on the interval $$ \left[0,\frac{\pi}{3}\right] $$:

$$
f_{\text{avg}} = \frac{1}{\frac{\pi}{3}-0} \int_0^{\frac{\pi}{3}} \sec^2 x\, dx = \frac{3}{\pi}\int_0^{\frac{\pi}{3}} \sec^2 x\, dx.
$$

We know that

$$
\int \sec^2 x\, dx = \tan x,
$$

so

$$
\int_0^{\frac{\pi}{3}} \sec^2 x\, dx = \tan\left(\frac{\pi}{3}\right) - \tan(0) = \sqrt{3} - 0 = \sqrt{3}.
$$

Thus, the average value is

$$
f_{\text{avg}} = \frac{3\sqrt{3}}{\pi}.
$$

The problem asks for the value of $$ c $$ in the interval $$ \left[0,\frac{\pi}{3}\right] $$ such that

$$
f(c) = \sec^2 c = \frac{3\sqrt{3}}{\pi}.
$$

To solve for $$ c $$, set

$$
\sec^2 c = \frac{3\sqrt{3}}{\pi}.
$$

Taking the reciprocal, we have

$$
\cos^2 c = \frac{\pi}{3\sqrt{3}}.
$$

Taking the square root of both sides (and noting that $$ \cos c $$ is nonnegative for $$ c \in \left[0,\frac{\pi}{3}\right] $$),

$$
\cos c = \sqrt{\frac{\pi}{3\sqrt{3}}}.
$$

Thus,

$$
c = \arccos\left(\sqrt{\frac{\pi}{3\sqrt{3}}}\right).
$$

Now we approximate the value. First, compute the value inside the square root:

- Approximate $$ \pi \approx 3.1416 $$ and $$ \sqrt{3} \approx 1.732 $$.
- Then, $$ 3\sqrt{3} \approx 3 \times 1.732 = 5.196 $$.
- So, $$ \frac{\pi}{3\sqrt{3}} \approx \frac{3.1416}{5.196} \approx 0.6046 $$.
- Taking the square root gives $$ \sqrt{0.6046} \approx 0.777 $$.

Finally, compute

$$
c \approx \arccos(0.777).
$$

Using a calculator,

$$
c \approx 0.685 \text{ radians}.
$$

Thus, the value of $$ c $$ is approximately

$$
c \approx 0.685.
$$
The value of $$ c $$ is approximately 0.685.

Given the function defined on the interval
below, find the value of where
the average value.

on the interval

Round to the nearest thousandth.

Upstudy AI Solution

Answer

Solution

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Latest Calculus Questions

Given the function defined on the interval below, find the value of where the average value. on the interval Round to the nearest thousandth.

Upstudy AI Solution

Answer

Solution

Extra Insights

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Latest Calculus Questions

Given the function defined on the interval
below, find the value of where
the average value.

on the interval

Round to the nearest thousandth.