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 Brooks Powers. 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 Brooks Powers. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We are given
$$
f(x)=x^4
$$
on the interval $$[-2,1]$$. The average value $$ f_{\text{avg}} $$ of a continuous function on $$[a,b]$$ is
$$
f_{\text{avg}}=\frac{1}{b-a}\int_{a}^{b}f(x)\,dx.
$$

**Step 1. Compute the Average Value**

Here, $$ a=-2 $$ and $$ b=1 $$, so
$$
f_{\text{avg}}=\frac{1}{1-(-2)}\int_{-2}^{1}x^4\,dx=\frac{1}{3}\int_{-2}^{1}x^4\,dx.
$$

**Step 2. Evaluate the Integral**

The antiderivative of $$ x^4 $$ is
$$
\frac{x^5}{5}.
$$
Evaluating from $$-2$$ to $$1$$,
$$
\int_{-2}^{1}x^4\,dx=\left[\frac{x^5}{5}\right]_{-2}^{1}=\frac{1^5}{5}-\frac{(-2)^5}{5}=\frac{1}{5}-\frac{-32}{5}=\frac{1+32}{5}=\frac{33}{5}.
$$

Thus, the average value is
$$
f_{\text{avg}}=\frac{1}{3}\cdot\frac{33}{5}=\frac{33}{15}=\frac{11}{5}=2.2.
$$

**Step 3. Find $$ c $$ Such That $$ f(c)=f_{\text{avg}} $$**

We need $$ c $$ satisfying
$$
f(c)=c^4=2.2.
$$
Taking the fourth root of both sides,
$$
c=\sqrt[4]{2.2}.
$$

**Step 4. Calculate $$ c $$ Numerically**

We compute:
$$
c\approx2.2^{1/4}\approx1.218.
$$

Thus, the value of $$ c $$ (rounded to the nearest thousandth) is
$$
\boxed{1.218}.
$$
The value of $$ c $$ is approximately 1.218.

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

The Deep Dive

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

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Given the function defined on the interval
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on the interval

Round to the nearest thousandth.