Average Value of Functions on Intervals - AP Calculus BC
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What is the average value of $f(x) = 8 - x^2$ on $[-2, 2]$?
What is the average value of $f(x) = 8 - x^2$ on $[-2, 2]$?
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$6$. Symmetric function $8 - x^2$ on symmetric interval $[-2,2]$.
$6$. Symmetric function $8 - x^2$ on symmetric interval $[-2,2]$.
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What is the average value of $f(x) = \text{ln}(x)$ on $[1, 2]$?
What is the average value of $f(x) = \text{ln}(x)$ on $[1, 2]$?
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$\text{ln}(2) - \frac{1}{2}$. Logarithm integrated by parts over interval $[1,2]$.
$\text{ln}(2) - \frac{1}{2}$. Logarithm integrated by parts over interval $[1,2]$.
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Calculate the average value of $f(x) = \sin(2x)$ on $[0, \frac{\pi}{2}]$.
Calculate the average value of $f(x) = \sin(2x)$ on $[0, \frac{\pi}{2}]$.
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$\frac{2}{\pi}$. Double angle sine function integrated over quarter period.
$\frac{2}{\pi}$. Double angle sine function integrated over quarter period.
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Find the average value of $f(x) = \frac{1}{x}$ on $[2, 4]$.
Find the average value of $f(x) = \frac{1}{x}$ on $[2, 4]$.
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$\frac{\text{ln}(2)}{2}$. Reciprocal function integrated over interval $[2,4]$ of length $2$.
$\frac{\text{ln}(2)}{2}$. Reciprocal function integrated over interval $[2,4]$ of length $2$.
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Find the average value of $f(x) = 4x^3$ on $[0, 1]$.
Find the average value of $f(x) = 4x^3$ on $[0, 1]$.
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$1$. Integrate $4x^3$ from $0$ to $1$ and divide by interval length.
$1$. Integrate $4x^3$ from $0$ to $1$ and divide by interval length.
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Find the average value of $f(x) = \ln(x)$ on $[1, e]$
Find the average value of $f(x) = \ln(x)$ on $[1, e]$
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$1 - \frac{1}{e}$. Natural logarithm integrated by parts over $[1,e]$
$1 - \frac{1}{e}$. Natural logarithm integrated by parts over $[1,e]$
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For $f(x) = e^x$, what is the average value on $[0, 1]$?
For $f(x) = e^x$, what is the average value on $[0, 1]$?
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$\frac{e - 1}{1}$. Exponential function integrated gives $e - 1$ over unit interval.
$\frac{e - 1}{1}$. Exponential function integrated gives $e - 1$ over unit interval.
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Which integral represents the average value of $f(x)$ on $[a, b]$?
Which integral represents the average value of $f(x)$ on $[a, b]$?
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$\frac{1}{b-a} \times \int_a^b f(x) , dx$. The fundamental formula for computing average value of any function.
$\frac{1}{b-a} \times \int_a^b f(x) , dx$. The fundamental formula for computing average value of any function.
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What is the average value of $f(x) = x^4 - 2x^2$ on $[-1, 1]$?
What is the average value of $f(x) = x^4 - 2x^2$ on $[-1, 1]$?
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$ -\frac{2}{5} $. Even function with negative quadratic term on symmetric interval.
$ -\frac{2}{5} $. Even function with negative quadratic term on symmetric interval.
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Which theorem relates to the average value of continuous functions?
Which theorem relates to the average value of continuous functions?
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Mean Value Theorem for Integrals. Guarantees existence of a point where function equals its average value.
Mean Value Theorem for Integrals. Guarantees existence of a point where function equals its average value.
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What is the average value of $f(x) = x^3$ on $[-1, 1]$?
What is the average value of $f(x) = x^3$ on $[-1, 1]$?
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$0$. Odd function $x^3$ on symmetric interval has zero average.
$0$. Odd function $x^3$ on symmetric interval has zero average.
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Calculate the average value of $f(x) = \cos x$ on $[0, \pi]$.
Calculate the average value of $f(x) = \cos x$ on $[0, \pi]$.
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$0$. Cosine function completes half cycle from $0$ to $\pi$.
$0$. Cosine function completes half cycle from $0$ to $\pi$.
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What is the average value of $f(x) = \text{e}^{2x}$ on $[0, 1]$?
What is the average value of $f(x) = \text{e}^{2x}$ on $[0, 1]$?
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$\frac{\text{e}^2 - 1}{2}$. Double exponential function $e^{2x}$ integrated over unit interval.
$\frac{\text{e}^2 - 1}{2}$. Double exponential function $e^{2x}$ integrated over unit interval.
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Identify the average value of $f(x) = \text{e}^x$ on $[1, 3]$.
Identify the average value of $f(x) = \text{e}^x$ on $[1, 3]$.
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$\frac{\text{e}^3 - \text{e}}{2}$. Exponential function over interval $[1,3]$ of length $2$.
$\frac{\text{e}^3 - \text{e}}{2}$. Exponential function over interval $[1,3]$ of length $2$.
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What is the average value of $f(x) = \frac{1}{x}$ on $[1, \text{e}]$?
What is the average value of $f(x) = \frac{1}{x}$ on $[1, \text{e}]$?
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$1$. Natural logarithm of $x$ gives $\ln(e) - \ln(1) = 1$.
$1$. Natural logarithm of $x$ gives $\ln(e) - \ln(1) = 1$.
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What does the average value formula calculate?
What does the average value formula calculate?
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The mean height of the function over an interval. Represents the constant height that gives same area as function.
The mean height of the function over an interval. Represents the constant height that gives same area as function.
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Determine the average value of $f(x) = \frac{1}{x^3}$ on $[1, 2]$.
Determine the average value of $f(x) = \frac{1}{x^3}$ on $[1, 2]$.
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$\frac{7}{24}$. Cubic negative power function integrated from $1$ to $2$.
$\frac{7}{24}$. Cubic negative power function integrated from $1$ to $2$.
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What is the average value of $f(x) = x^4$ on $[0, 2]$?
What is the average value of $f(x) = x^4$ on $[0, 2]$?
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$\frac{16}{5}$. Fourth power function $x^4$ integrated from $0$ to $2$.
$\frac{16}{5}$. Fourth power function $x^4$ integrated from $0$ to $2$.
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What is the average value of $f(x) = x^2$ on $[0, 2]$?
What is the average value of $f(x) = x^2$ on $[0, 2]$?
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$\frac{2}{3}$. Integrate $x^2$ from $0$ to $2$, then divide by interval length $2$.
$\frac{2}{3}$. Integrate $x^2$ from $0$ to $2$, then divide by interval length $2$.
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For $f(x) = x$, find the average value on $[1, 4]$.
For $f(x) = x$, find the average value on $[1, 4]$.
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$2.5$. Linear function $x$ has average equal to midpoint of interval.
$2.5$. Linear function $x$ has average equal to midpoint of interval.
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Determine the average value of $f(x) = x^2 + 1$ on $[0, 3]$.
Determine the average value of $f(x) = x^2 + 1$ on $[0, 3]$.
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$4$. Quadratic plus constant function integrated over $[0,3]$.
$4$. Quadratic plus constant function integrated over $[0,3]$.
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What is the average value of $f(x) = 3x^2 - 2x$ on $[0, 1]$?
What is the average value of $f(x) = 3x^2 - 2x$ on $[0, 1]$?
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$\frac{4}{3}$. Cubic minus quadratic function integrated over unit interval.
$\frac{4}{3}$. Cubic minus quadratic function integrated over unit interval.
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For $f(x) = \text{sin}^2(x)$, what is the average value on $[0, \text{pi}]$?
For $f(x) = \text{sin}^2(x)$, what is the average value on $[0, \text{pi}]$?
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$\frac{1}{2}$. Squared sine uses identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$.
$\frac{1}{2}$. Squared sine uses identity $\sin^2 x = \frac{1 - \cos(2x)}{2}$.
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Identify the average value of $f(x) = \text{cos}^2(x)$ on $[0, \text{pi}]$.
Identify the average value of $f(x) = \text{cos}^2(x)$ on $[0, \text{pi}]$.
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$\frac{1}{2}$. Squared cosine uses identity $\cos^2 x = \frac{1 + \cos(2x)}{2}$.
$\frac{1}{2}$. Squared cosine uses identity $\cos^2 x = \frac{1 + \cos(2x)}{2}$.
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Determine the average value of $f(x) = \text{tan}(x)$ on $[0, \frac{\text{pi}}{4}]$.
Determine the average value of $f(x) = \text{tan}(x)$ on $[0, \frac{\text{pi}}{4}]$.
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$\text{ln}(2)$. Tangent function integrated gives $-\ln(\cos x)$ from $0$ to $\frac{\pi}{4}$.
$\text{ln}(2)$. Tangent function integrated gives $-\ln(\cos x)$ from $0$ to $\frac{\pi}{4}$.
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State the formula for the average value of a function $f(x)$ on $[a, b]$.
State the formula for the average value of a function $f(x)$ on $[a, b]$.
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$\frac{1}{b-a} \times \int_a^b f(x) , dx$. Uses the integral divided by interval length to find mean height.
$\frac{1}{b-a} \times \int_a^b f(x) , dx$. Uses the integral divided by interval length to find mean height.
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Calculate the average value of $f(x) = \frac{1}{x}$ on $[1, 4]$.
Calculate the average value of $f(x) = \frac{1}{x}$ on $[1, 4]$.
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$\frac{\ln(4)}{3}$. Integrate $\frac{1}{x}$ to get $\ln(x)$, evaluate from $1$ to $4$.
$\frac{\ln(4)}{3}$. Integrate $\frac{1}{x}$ to get $\ln(x)$, evaluate from $1$ to $4$.
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What is the average value of $f(x) = x^2 - x$ on $[0, 3]$?
What is the average value of $f(x) = x^2 - x$ on $[0, 3]$?
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$2$. Quadratic minus linear term integrated over $[0,3]$.
$2$. Quadratic minus linear term integrated over $[0,3]$.
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What is the average value of $f(x) = 2x$ on $[0, 3]$?
What is the average value of $f(x) = 2x$ on $[0, 3]$?
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$3$. Linear function $2x$ integrated from $0$ to $3$ gives average $3$.
$3$. Linear function $2x$ integrated from $0$ to $3$ gives average $3$.
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Identify the average value of $f(x) = \text{sin}x$ from $0$ to $\frac{\text{pi}}{2}$.
Identify the average value of $f(x) = \text{sin}x$ from $0$ to $\frac{\text{pi}}{2}$.
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$\frac{2}{\text{pi}}$. Integrate $\sin x$ from $0$ to $\frac{\pi}{2}$ and divide by $\frac{\pi}{2}$.
$\frac{2}{\text{pi}}$. Integrate $\sin x$ from $0$ to $\frac{\pi}{2}$ and divide by $\frac{\pi}{2}$.
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