Difference between revisions of "009B Sample Midterm 3, Problem 2"

Revision as of 09:48, 6 February 2017

State the fundamental theorem of calculus, and use this theorem to find the derivative of

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $G(x)=\int _{x}^{5}{\frac {1}{1+u^{10}}}~du?$
First, we need to switch the bounds of integration.
So, we have $G(x)=-\int _{5}^{x}{\frac {1}{1+u^{10}}}~du.$
By Part 1 of the Fundamental Theorem of Calculus, $G'(x)=-{\frac {1}{1+x^{10}}}.$

Solution:

Step 1:
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ and let $F(x)=\int _{a}^{x}f(t)~dt.$
Then, $F$ is a differentiable function on $(a,b)$ and $F'(x)=f(x).$
The Fundamental Theorem of Calculus, Part 2
Let $f$ be continuous on $[a,b]$ and let $F$ be any antiderivative of $f.$
Then, $\int _{a}^{b}f(x)~dx=F(b)-F(a).$

Step 2:
First, we have
$F(x)=-\int _{5}^{\cos(x)}{\frac {1}{1+u^{10}}}~du.$
Now, let $g(x)=\cos(x)$ and $G(x)=\int _{5}^{x}{\frac {1}{1+u^{10}}}~du.$
So,
$F(x)=-G(g(x)).$
Hence, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F'(x)=-G'(g(x))g'(x)} by the Chain Rule.

Step 3:
Now, $g'(x)=-\sin(x).$
By the Fundamental Theorem of Calculus,
$G'(x)={\frac {1}{1+x^{10}}}.$
Hence,
${\begin{array}{rcl}\displaystyle {F'(x)}&=&\displaystyle {-{\frac {1}{1+\cos ^{10}x}}(-\sin(x))}\\&&\\&=&\displaystyle {{\frac {\sin(x)}{1+\cos ^{10}x}}.}\\\end{array}}$

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 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).}
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}}

Return to Sample Exam

@@ Line 18: / Line 18: @@
 ::By Part 1 of the Fundamental Theorem of Calculus, <math style="vertical-align: -16px">G'(x)=-\frac{1}{1+x^{10}}.</math>
 |}
 '''Solution:'''
@@ Line 77: / Line 78: @@
 \end{array}</math>
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''The Fundamental Theorem of Calculus, Part 1'''
+|'''The Fundamental Theorem of Calculus, Part 1'''
 |-
 |
-:&nbsp;&nbsp; Let <math style="vertical-align: -5px">f</math> be continuous on <math style="vertical-align: -5px">[a,b]</math> and let <math style="vertical-align: -14px">F(x)=\int_a^x f(t)~dt.</math>
+Let <math style="vertical-align: -5px">f</math> be continuous on <math style="vertical-align: -5px">[a,b]</math> and let <math style="vertical-align: -14px">F(x)=\int_a^x f(t)~dt.</math>
 |-
 |
-:&nbsp;&nbsp; Then, <math style="vertical-align: -1px">F</math> is a differentiable function on <math style="vertical-align: -5px">(a,b)</math> and <math style="vertical-align: -5px">F'(x)=f(x).</math>
+Then, <math style="vertical-align: -1px">F</math> is a differentiable function on <math style="vertical-align: -5px">(a,b)</math> and <math style="vertical-align: -5px">F'(x)=f(x).</math>
 |-
-|&nbsp;&nbsp; '''The Fundamental Theorem of Calculus, Part 2'''
+|'''The Fundamental Theorem of Calculus, Part 2'''
 |-
 |
-:&nbsp;&nbsp; Let <math style="vertical-align: -5px">f</math> be continuous on <math style="vertical-align: -5px">[a,b]</math> and let <math style="vertical-align: -1px">F</math> be any antiderivative of <math style="vertical-align: -5px">f.</math>
+Let <math style="vertical-align: -5px">f</math> be continuous on <math style="vertical-align: -5px">[a,b]</math> and let <math style="vertical-align: -1px">F</math> be any antiderivative of <math style="vertical-align: -5px">f.</math>
 |-
 |
-:&nbsp;&nbsp; Then, <math style="vertical-align: -14px">\int_a^b f(x)~dx=F(b)-F(a).</math>
+Then, <math style="vertical-align: -14px">\int_a^b f(x)~dx=F(b)-F(a).</math>
 |-
 |&nbsp;&nbsp; <math>F'(x)=\frac{\sin(x)}{1+\cos^{10}x}</math>
 |}
 [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009B Sample Midterm 3, Problem 2"

Revision as of 09:48, 6 February 2017

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