Difference between revisions of "009B Sample Midterm 2, Problem 2"
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::<span class="exam">a) State the fundamental theorem of calculus. | ::<span class="exam">a) State the fundamental theorem of calculus. | ||
| − | ::<span class="exam">b) Compute <math>\frac{d}{dx}\int_0^{\cos (x)}\sin (t)dt</math> | + | ::<span class="exam">b) Compute <math>\frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt</math> |
| − | ::<span class="exam">c) Evaluate <math>\int_{0}^{\frac{\pi}{4}}\sec^2 | + | ::<span class="exam">c) Evaluate <math>\int_{0}^{\frac{\pi}{4}}\sec^2 x~dx</math> |
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|'''The Fundamental Theorem of Calculus, Part 1''' | |'''The Fundamental Theorem of Calculus, Part 1''' | ||
|- | |- | ||
| − | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F(x)=\int_a^x f(t)dt</math>. | + | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F(x)=\int_a^x f(t)~dt</math>. |
|- | |- | ||
|Then, <math>F</math> is a differential function on <math>(a,b)</math> and <math>F'(x)=f(x)</math>. | |Then, <math>F</math> is a differential function on <math>(a,b)</math> and <math>F'(x)=f(x)</math>. | ||
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|Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F</math> be any antiderivative of <math>f</math>. | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F</math> be any antiderivative of <math>f</math>. | ||
|- | |- | ||
| − | |Then, <math>\int_a^b f(x)dx=F(b)-F(a)</math> | + | |Then, <math>\int_a^b f(x)~dx=F(b)-F(a)</math> |
|} | |} | ||
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!Step 1: | !Step 1: | ||
|- | |- | ||
| − | |Let <math>F(x)=\int_0^{\cos (x)}\sin (t)dt</math>. The problem is asking us to find <math>F'(x)</math>. | + | |Let <math>F(x)=\int_0^{\cos (x)}\sin (t)~dt</math>. The problem is asking us to find <math>F'(x)</math>. |
|- | |- | ||
| − | |Let <math>g(x)=\cos(x)</math> and <math>G(x)=\int_0^x \sin(t)dt</math>. | + | |Let <math>g(x)=\cos(x)</math> and <math>G(x)=\int_0^x \sin(t)~dt</math>. |
|- | |- | ||
|Then, <math>F(x)=G(g(x))</math>. | |Then, <math>F(x)=G(g(x))</math>. | ||
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| Using the '''Fundamental Theorem of Calculus, Part 2''', we have | | Using the '''Fundamental Theorem of Calculus, Part 2''', we have | ||
|- | |- | ||
| − | |<math>\int_{0}^{\frac{\pi}{4}}\sec^2 | + | |<math>\int_{0}^{\frac{\pi}{4}}\sec^2 x~dx=\left.\tan(x)\right|_0^{\frac{\pi}{4}}</math> |
|} | |} | ||
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|So, we get | |So, we get | ||
|- | |- | ||
| − | |<math>\int_{0}^{\frac{\pi}{4}}\sec^2 | + | |<math>\int_{0}^{\frac{\pi}{4}}\sec^2 x~dx=\tan \bigg(\frac{\pi}{4}\bigg)-\tan (0)=1</math> |
|- | |- | ||
| | | | ||
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|'''The Fundamental Theorem of Calculus, Part 1''' | |'''The Fundamental Theorem of Calculus, Part 1''' | ||
|- | |- | ||
| − | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F(x)=\int_a^x f(t)dt</math>. | + | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F(x)=\int_a^x f(t)~dt</math>. |
|- | |- | ||
|Then, <math>F</math> is a differential function on <math>(a,b)</math> and <math>F'(x)=f(x)</math>. | |Then, <math>F</math> is a differential function on <math>(a,b)</math> and <math>F'(x)=f(x)</math>. | ||
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|Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F</math> be any antiderivative of <math>f</math>. | |Let <math>f</math> be continuous on <math>[a,b]</math> and let <math>F</math> be any antiderivative of <math>f</math>. | ||
|- | |- | ||
| − | |Then, <math>\int_a^b f(x)dx=F(b)-F(a)</math> | + | |Then, <math>\int_a^b f(x)~dx=F(b)-F(a)</math> |
|- | |- | ||
|'''(b)''' <math>\sin(\cos(x))(-\sin(x)</math> | |'''(b)''' <math>\sin(\cos(x))(-\sin(x)</math> | ||
Revision as of 14:19, 31 January 2016
This problem has three parts:
- a) State the fundamental theorem of calculus.
- b) Compute 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://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt}
- c) Evaluate 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://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\frac{\pi}{4}}\sec^2 x~dx}
| Foundations: |
|---|
Solution:
(a)
| Step 1: |
|---|
| The Fundamental Theorem of Calculus has two parts. |
| The Fundamental Theorem of Calculus, Part 1 |
| Let 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://wikimedia.org/api/rest_v1/":): {\displaystyle f} be continuous on 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://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and let 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://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=\int_a^x f(t)~dt} . |
| Then, 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://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a differential function on 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://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)} and 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://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x)=f(x)} . |
| Step 2: |
|---|
| The Fundamental Theorem of Calculus, Part 2 |
| Let 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://wikimedia.org/api/rest_v1/":): {\displaystyle f} be continuous on 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://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and let be any antiderivative of . |
| Then, |
(b)
| Step 1: |
|---|
| Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x)=\int _{0}^{\cos(x)}\sin(t)~dt} . The problem is asking us to find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F'(x)} . |
| Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=\cos(x)} and 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://wikimedia.org/api/rest_v1/":): {\displaystyle G(x)=\int_0^x \sin(t)~dt} . |
| Then, 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://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=G(g(x))} . |
| Step 2: |
|---|
| If we take the derivative of both sides of the last equation, we get 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://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x)=G'(g(x))g'(x)} by the Chain Rule. |
| Step 3: |
|---|
| Now, 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://wikimedia.org/api/rest_v1/":): {\displaystyle g'(x)=-\sin(x)} and 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://wikimedia.org/api/rest_v1/":): {\displaystyle G'(x)=\sin(x)} by the Fundamental Theorem of Calculus, Part 1. |
| Since 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://wikimedia.org/api/rest_v1/":): {\displaystyle G'(g(x))=\sin(g(x))=\sin(\cos(x))} , we have 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://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x)=G'(g(x))g'(x)=\sin(\cos(x))(-\sin(x))} |
(c)
| Step 1: |
|---|
| Using the Fundamental Theorem of Calculus, Part 2, we have |
| 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://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\frac{\pi}{4}}\sec^2 x~dx=\left.\tan(x)\right|_0^{\frac{\pi}{4}}} |
| Step 2: |
|---|
| So, we get |
| 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://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{\frac{\pi}{4}}\sec^2 x~dx=\tan \bigg(\frac{\pi}{4}\bigg)-\tan (0)=1} |
| Final Answer: |
|---|
| (a) |
| The Fundamental Theorem of Calculus, Part 1 |
| Let 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://wikimedia.org/api/rest_v1/":): {\displaystyle f} be continuous on 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://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and let 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://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)=\int_a^x f(t)~dt} . |
| Then, 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://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a differential function on 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://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)} and 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://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x)=f(x)} . |
| The Fundamental Theorem of Calculus, Part 2 |
| Let 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://wikimedia.org/api/rest_v1/":): {\displaystyle f} be continuous on 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://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} and let 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://wikimedia.org/api/rest_v1/":): {\displaystyle F} be any antiderivative of 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://wikimedia.org/api/rest_v1/":): {\displaystyle f} . |
| Then, 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://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b f(x)~dx=F(b)-F(a)} |
| (b) 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://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\cos(x))(-\sin(x)} |
| (c) 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://wikimedia.org/api/rest_v1/":): {\displaystyle 1} |