6.5 Average Value of a Function/2: Difference between revisions
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<math> | <math> | ||
f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi | f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] | ||
</math> | </math> | ||
<math> | <math> | ||
| Line 8: | Line 9: | ||
f_{avg} &= \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx \\[2ex] | f_{avg} &= \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx \\[2ex] | ||
&= \frac{1}{2\pi}\int_{-4\pi}^{4\pi}\sin{(u)}\frac{1}{4}\,du = \frac{1}{8\pi}\int_{-4\pi}^{4\pi}\sin(u)\,du \\[2ex] | &= \frac{1}{2\pi}\int_{-4\pi}^{4\pi}\sin{(u)}\left(\frac{1}{4}\,du\right) = \frac{1}{8\pi}\int_{-4\pi}^{4\pi}\sin(u)\,du \\[2ex] | ||
&= -\frac{1}{8\pi}\cos(u)\bigg|_{-4\pi}^{4\pi} \\[2ex] | &= -\frac{1}{8\pi}\cos(u)\bigg|_{-4\pi}^{4\pi} \\[2ex] | ||
&= \left[-\frac{1}{8\pi}\cos(4\pi)\right]-\left[-\frac{1}{8\pi}\cos(-4\pi)\right] | &= \left[-\frac{1}{8\pi}\cos(4\pi)\right]-\left[-\frac{1}{8\pi}\cos(-4\pi)\right] = \left[-\frac{1}{8\pi}(1)\right]+\left[\frac{1}{8\pi}(1)\right] \\[2ex] | ||
&= 0 | |||
\end{align} | \end{align} | ||
</math> | </math> | ||
U-Sub notes:<br> | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
Latest revision as of 18:10, 7 September 2022
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(x) = \sin{(4x)}\text{,}\quad [-\pi, \pi] }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{avg} &= \frac{1}{\pi-(-\pi)}\int_{-\pi}^{\pi}\sin{(4x)}\,dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}\sin{(4x)}\,dx \\[2ex] &= \frac{1}{2\pi}\int_{-4\pi}^{4\pi}\sin{(u)}\left(\frac{1}{4}\,du\right) = \frac{1}{8\pi}\int_{-4\pi}^{4\pi}\sin(u)\,du \\[2ex] &= -\frac{1}{8\pi}\cos(u)\bigg|_{-4\pi}^{4\pi} \\[2ex] &= \left[-\frac{1}{8\pi}\cos(4\pi)\right]-\left[-\frac{1}{8\pi}\cos(-4\pi)\right] = \left[-\frac{1}{8\pi}(1)\right]+\left[\frac{1}{8\pi}(1)\right] \\[2ex] &= 0 \end{align} }
U-Sub notes:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u &=4x \\[2ex] du &= 4dx \\[2ex] \frac{1}{4}du &= dx \end{align} }
New upper limit: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4\pi = 4(\pi)}
New lower limit: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -4\pi = 4(-\pi)}