Difference between revisions of "009B Sample Midterm 3, Problem 2"
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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 differentiable function on <math>(a,b)</math> and <math>F'(x)=f(x)</math> | + | |Then, <math>F</math> is a differentiable function on <math>(a,b)</math> and <math>F'(x)=f(x).</math> |
|- | |- | ||
|'''The Fundamental Theorem of Calculus, Part 2''' | |'''The Fundamental Theorem of Calculus, Part 2''' | ||
|- | |- | ||
| − | |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 2: | !Step 2: | ||
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| − | |First, we have <math>F(x)=-\int_5^{\cos (x)} \frac{1}{1+u^{10}}~du</math> | + | |First, we have <math>F(x)=-\int_5^{\cos (x)} \frac{1}{1+u^{10}}~du.</math> |
|- | |- | ||
| − | |Now, let <math>g(x)=\cos(x)</math> and <math>G(x)=\int_5^x \frac{1}{1+u^{10}}~du</math> | + | |Now, let <math>g(x)=\cos(x)</math> and <math>G(x)=\int_5^x \frac{1}{1+u^{10}}~du.</math> |
|- | |- | ||
| − | |So, <math>F(x)=-G(g(x))</math> | + | |So, <math>F(x)=-G(g(x)).</math> |
|- | |- | ||
|Hence, <math>F'(x)=-G'(g(x))g'(x)</math> by the Chain Rule. | |Hence, <math>F'(x)=-G'(g(x))g'(x)</math> by the Chain Rule. | ||
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!Step 3: | !Step 3: | ||
|- | |- | ||
| − | |Now, <math>g'(x)=-\sin(x)</math> | + | |Now, <math>g'(x)=-\sin(x).</math> |
|- | |- | ||
| − | | By the Fundamental Theorem of Calculus, <math>G'(x)=\frac{1}{1+x^{10}}</math> | + | | By the Fundamental Theorem of Calculus, <math>G'(x)=\frac{1}{1+x^{10}}.</math> |
|- | |- | ||
| − | |Hence, <math>F'(x)=-\frac{1}{1+\cos^{10}x}(-\sin(x))=\frac{\sin(x)}{1+\cos^{10}x}</math> | + | |Hence, <math>F'(x)=-\frac{1}{1+\cos^{10}x}(-\sin(x))=\frac{\sin(x)}{1+\cos^{10}x}.</math> |
|- | |- | ||
| | | | ||
| Line 67: | Line 67: | ||
|'''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 differentiable function on <math>(a,b)</math> and <math>F'(x)=f(x)</math> | + | |Then, <math>F</math> is a differentiable function on <math>(a,b)</math> and <math>F'(x)=f(x).</math> |
|- | |- | ||
|'''The Fundamental Theorem of Calculus, Part 2''' | |'''The Fundamental Theorem of Calculus, Part 2''' | ||
|- | |- | ||
| − | |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> |
|- | |- | ||
| − | | <math>F'(x)=\frac{\sin(x)}{1+\cos^{10}x}</math> | + | |<math>F'(x)=\frac{\sin(x)}{1+\cos^{10}x}</math> |
|} | |} | ||
[[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 15:38, 29 March 2016
State the fundamental theorem of calculus, and use this theorem to find the derivative 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(x)=\int_{\cos (x)}^5 \frac{1}{1+u^{10}}~du.}
| Foundations: |
|---|
| What does Part 1 of the Fundamental Theorem of Calculus say is the derivative 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 G(x)=\int_x^5 \frac{1}{1+u^{10}}~du?} |
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Solution:
| 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 differentiable 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).} |
| Step 2: |
|---|
| First, 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)=-\int_5^{\cos (x)} \frac{1}{1+u^{10}}~du.} |
| Now, 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 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_5^x \frac{1}{1+u^{10}}~du.} |
| So, 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)).} |
| Hence, 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).} |
| By the Fundamental Theorem of Calculus, 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)=\frac{1}{1+x^{10}}.} |
| Hence, 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)=-\frac{1}{1+\cos^{10}x}(-\sin(x))=\frac{\sin(x)}{1+\cos^{10}x}.} |
| Final Answer: |
|---|
| 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 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 differentiable 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).} |
| 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)=\frac{\sin(x)}{1+\cos^{10}x}} |
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)