Difference between revisions of "009B Sample Midterm 1, Problem 2"

Revision as of 15:25, 26 January 2016

Find the average value of the function on the given interval.

f(x)=2x^{3}(1+x^{2})^{4},~~~[0,2]

Foundations:
The average value of a function $f(x)$ on an interval $[a,b]$ is given by $f_{\text{avg}}={\frac {1}{b-a}}\int _{a}^{b}f(x)dx$ .

Solution:

Step 1:
Using the formula given in the Foundations sections, we have:
$f_{\text{avg}}={\frac {1}{2-0}}\int _{0}^{2}2x^{3}(1+x^{2})^{4}dx=\int _{0}^{2}x^{3}(1+x^{2})^{4}dx$

Final Answer:

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+|The average value of a function <math>f(x)</math> on an interval <math>[a,b]</math> is given by <math>f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)dx</math>.
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+|Using the formula given in the Foundations sections, we have:
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+|<math>f_{\text{avg}}=\frac{1}{2-0}\int_0^2 2x^3(1+x^2)^4dx=\int_0^2 x^3(1+x^2)^4dx</math>
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