5.3 The Fundamental Theorem of Calculus/15: Difference between revisions
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Use part 1 of the FTC to find the derivative of the function: | Use part 1 of the FTC to find the derivative of the function: | ||
<math>y=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt</math> | <math>y=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt</math> | ||
FTC 1: | |||
<math>\frac{d}{dx}\int_{a(x)}^{b(x)}f(t)\,dt=b^\prime{(x)}\cdot\,f(b(x))-\,a^\prime{(x)}\cdot\,f(a(x))</math> | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\frac{d}{dx}=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt =\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})-0\cdot\sqrt{0+\sqrt 0}=\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)}) | \frac{d}{dx}=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt | ||
&=\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})-0\cdot\sqrt{0+\sqrt 0} | |||
&=\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)}) | |||
\end{align} | \end{align} | ||
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In this problem <math>a^\prime{(x)}= 0</math>, so when it is multiplied by <math>f(a(x))</math> it will result in 0 and doesn't need to be added. | In this problem <math>a^\prime{(x)}= 0</math>, so when it is multiplied by <math>f(a(x))</math> it will result in 0 and doesn't need to be added. | ||
Revision as of 19:52, 25 August 2022
Use part 1 of the FTC to find the derivative of the function: 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 y=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt}
FTC 1: 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 \frac{d}{dx}\int_{a(x)}^{b(x)}f(t)\,dt=b^\prime{(x)}\cdot\,f(b(x))-\,a^\prime{(x)}\cdot\,f(a(x))}
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} \frac{d}{dx}=\int_{0}^{tan(x)}\sqrt{t+\sqrt t}\,dt &=\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)})-0\cdot\sqrt{0+\sqrt 0} &=\sec^{2}(x)\cdot\sqrt{tan(x)+\sqrt tan(x)}) \end{align} }
In this problem 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 a^\prime{(x)}= 0}
, so when it is multiplied by 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(a(x))}
it will result in 0 and doesn't need to be added.