Average Value of a Function on an Interval Practice

Sample questions

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1. Sample 1difficulty 1/5

   The average value of $f$ on $[a, b]$ is

   • A

     $f((a+b)/2)$

   • B

     $\int_a^b f\,dx$

   • C

     $\dfrac{1}{b-a}\int_a^b f\,dx$

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   • D

     $\dfrac{f(a) + f(b)}{2}$

   Why

   Standard definition.

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2. Sample 2difficulty 2/5

   The "average value of $f$ on $[a, b]$" is <strong>the same as</strong>

   • A

     $f(b) - f(a)$

   • B

     Slope of secant from $(a, f(a))$ to $(b, f(b))$

   • C

     Average rate of change of $f$

   • D

     $1/(b-a) \int_a^b f$

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   Why

   Definition.

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3. Sample 3difficulty 2/5

   Average of $\sin x$ on $[0, \pi]$ is

   • A

     $0$

   • B

     $\tfrac{2}{\pi}$

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   • C

     $1$

   • D

     $\tfrac{1}{\pi}$

   Why

   $\dfrac{1}{\pi}\int_0^\pi \sin x\,dx = (1/\pi)(2) = 2/\pi$.

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4. Sample 4difficulty 2/5

   Average value of $e^x$ on $[0, 1]$:

   • A

     $e - 1$

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   • B

     $\dfrac{e + 1}{2}$

   • C

     $e/2$

   • D

     $1$

   Why

   $\int_0^1 e^x\,dx/(1) = e - 1$.

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5. Sample 5difficulty 2/5

   Average value of $f(x) = x^2$ on $[0, 3]$ is

   • A

     $3$

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   • B

     $\tfrac{9}{2}$

   • C

     $1$

   • D

     $3$

   Why

   $\dfrac{1}{3}\int_0^3 x^2\,dx = \dfrac{1}{3} \cdot 9 = 3$.

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