Question

(1 point) find the average value of the function ( f(x) = sqrt3{x} ) on the interval 1, 27. ( f_{\text{ave}} = ) you have attempted this problem 0 times. you have unlimited attempts remaining. preview my answers submit answers

Explanation:

Step1: Recall the average value formula

The average value of a function \( f(x) \) on the interval \([a, b]\) is given by \( f_{\text{ave}}=\frac{1}{b - a}\int_{a}^{b}f(x)dx \). Here, \( a = 1 \), \( b = 27 \), and \( f(x)=\sqrt[3]{x}=x^{\frac{1}{3}} \).

Step2: Compute the integral

First, find the antiderivative of \( x^{\frac{1}{3}} \). Using the power rule for integration \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C \) (where \( n eq - 1 \)), for \( n=\frac{1}{3} \), the antiderivative \( F(x)=\frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}=\frac{x^{\frac{4}{3}}}{\frac{4}{3}}=\frac{3}{4}x^{\frac{4}{3}} \).
Then, evaluate the definite integral \( \int_{1}^{27}x^{\frac{1}{3}}dx=F(27)-F(1) \).
Calculate \( F(27)=\frac{3}{4}(27)^{\frac{4}{3}} \). Since \( 27 = 3^{3} \), \( (27)^{\frac{4}{3}}=(3^{3})^{\frac{4}{3}}=3^{4}=81 \), so \( F(27)=\frac{3}{4}\times81=\frac{243}{4} \).
Calculate \( F(1)=\frac{3}{4}(1)^{\frac{4}{3}}=\frac{3}{4}\times1=\frac{3}{4} \).
So, \( \int_{1}^{27}x^{\frac{1}{3}}dx=\frac{243}{4}-\frac{3}{4}=\frac{240}{4} = 60 \).

Step3: Compute the average value

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Answer:

Step1: Recall the average value formula

The average value of a function \( f(x) \) on the interval \([a, b]\) is given by \( f_{\text{ave}}=\frac{1}{b - a}\int_{a}^{b}f(x)dx \). Here, \( a = 1 \), \( b = 27 \), and \( f(x)=\sqrt[3]{x}=x^{\frac{1}{3}} \).

Step2: Compute the integral

First, find the antiderivative of \( x^{\frac{1}{3}} \). Using the power rule for integration \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C \) (where \( n
eq - 1 \)), for \( n=\frac{1}{3} \), the antiderivative \( F(x)=\frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1}=\frac{x^{\frac{4}{3}}}{\frac{4}{3}}=\frac{3}{4}x^{\frac{4}{3}} \).
Then, evaluate the definite integral \( \int_{1}^{27}x^{\frac{1}{3}}dx=F(27)-F(1) \).
Calculate \( F(27)=\frac{3}{4}(27)^{\frac{4}{3}} \). Since \( 27 = 3^{3} \), \( (27)^{\frac{4}{3}}=(3^{3})^{\frac{4}{3}}=3^{4}=81 \), so \( F(27)=\frac{3}{4}\times81=\frac{243}{4} \).
Calculate \( F(1)=\frac{3}{4}(1)^{\frac{4}{3}}=\frac{3}{4}\times1=\frac{3}{4} \).
So, \( \int_{1}^{27}x^{\frac{1}{3}}dx=\frac{243}{4}-\frac{3}{4}=\frac{240}{4} = 60 \).

Step3: Compute the average value

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