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

Revision as of 11:11, 27 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 xdx}

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 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)

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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x)=\int _{0}^{x}\sin(t)dt} .
Then, $F(x)=G(g(x))$ .

Step 2:
If we take the derivative of both sides of the last equation, we get $F'(x)=G'(g(x))g'(x)$ by the Chain Rule.

Step 3:
Now, $g'(x)=-\sin(x)$ and $G'(x)=\sin(x)$ by the Fundamental Theorem of Calculus, Part 1.
Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G'(g(x))=\sin(g(x))=\sin(\cos(x))} , we have $F'(x)=G'(g(x))g'(x)=\sin(\cos(x))(-\sin(x))$

(c)

Step 1:
Using the Fundamental Theorem of Calculus, Part 2, we have
$\int _{0}^{\frac {\pi }{4}}\sec ^{2}xdx=\left.\tan(x)\right\|_{0}^{\frac {\pi }{4}}$

Step 2:
So, we get
$\int _{0}^{\frac {\pi }{4}}\sec ^{2}xdx=\tan {\bigg (}{\frac {\pi }{4}}{\bigg )}-\tan(0)=1$

Final Answer:
(a)
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ 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}

Return to Sample Exam

@@ Line 43: / Line 43: @@
 !Step 1: &nbsp;
 |-
-|
+|Let <math>F(x)=\int_0^{\cos (x)}\sin (t)dt</math>. The problem is asking us to find <math>F'(x)</math>.
 |-
-|'''The Fundamental Theorem of Calculus, Part 1'''
+|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>.
-|-
-|
 |}
@@ Line 55: / Line 53: @@
 !Step 2: &nbsp;
 |-
-|
+|If we take the derivative of both sides of the last equation, we get <math>F'(x)=G'(g(x))g'(x)</math> by the Chain Rule.
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+!Step 3: &nbsp;
 |-
-|
+|Now, <math>g'(x)=-\sin(x)</math> and <math>G'(x)=\sin(x)</math> by the '''Fundamental Theorem of Calculus, Part 1'''.
 |-
-|
+|Since <math>G'(g(x))=\sin(g(x))=\sin(\cos(x))</math>, we have <math>F'(x)=G'(g(x))g'(x)=\sin(\cos(x))(-\sin(x))</math>
 |}
@@ Line 99: / Line 101: @@
 |Then, <math>\int_a^b f(x)dx=F(b)-F(a)</math>
 |-
-|'''(b)'''
+|'''(b)''' <math>\sin(\cos(x))(-\sin(x)</math>
 |-
 |'''(c)''' <math>1</math>
 |}
 [[009B_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

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

Revision as of 11:11, 27 January 2016

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