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| | ::<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt</math> | | ::<math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt</math> |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| + | <hr> |
| − | !Foundations:
| + | [[009B Sample Final 2, Problem 1 Solution|'''<u>Solution</u>''']] |
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| − | |'''1.''' What does Part 2 of the Fundamental Theorem of Calculus say about <math style="vertical-align: -15px">\int_a^b\sec^2x~dx</math> where <math style="vertical-align: -5px">a,b</math> are constants? | |
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| − | Part 2 of the Fundamental Theorem of Calculus says that
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| − | | <math style="vertical-align: -15px">\int_a^b\sec^2x~dx=F(b)-F(a)</math> where <math style="vertical-align: 0px">F</math> is any antiderivative of <math style="vertical-align: 0px">\sec^2x.</math>
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| − | |'''2.''' What does Part 1 of the Fundamental Theorem of Calculus say about <math style="vertical-align: -15px">\frac{d}{dx}\int_0^x\sin(t)~dt?</math>
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| − | Part 1 of the Fundamental Theorem of Calculus says that
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| − | | <math style="vertical-align: -15px">\frac{d}{dx}\int_0^x\sin(t)~dt=\sin(x).</math>
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| − | '''Solution:'''
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| − | '''(a)''' | + | [[009B Sample Final 2, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |The Fundamental Theorem of Calculus has two parts.
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| − | |'''The Fundamental Theorem of Calculus, Part 1'''
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| − | | Let <math>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>
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| − | | Then, <math style="vertical-align: 0px">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>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |'''The Fundamental Theorem of Calculus, Part 2'''
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| − | | Let <math>f</math> be continuous on <math>[a,b]</math> and let <math style="vertical-align: 0px">F</math> be any antiderivative of <math>f.</math>
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| − | | Then, <math style="vertical-align: -14px">\int_a^b f(x)~dx=F(b)-F(a).</math>
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| − | '''(b)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |The Fundamental Theorem of Calculus Part 2 says that
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| − | | <math>\int_0^1 \frac{d}{dx}\bigg(e^{\arctan(x)}\bigg)~dx=F(1)-F(0)</math>
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| − | |where <math style="vertical-align: -5px">F(x)</math> is any antiderivative of <math style="vertical-align: -15px">\frac{d}{dx}\bigg(e^{\arctan(x)}\bigg).</math>
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| − | |Thus, we can take
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| − | | <math>F(x)=e^{\arctan(x)}</math>
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| − | |since then <math style="vertical-align: -15px">F'(x)=\frac{d}{dx}\bigg(e^{\arctan(x)}\bigg).</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Now, we have
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| − | | <math>\begin{array}{rcl}
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| − | \displaystyle{\int_0^1 \frac{d}{dx}\bigg(e^{\arctan(x)}\bigg)~dx} & = & \displaystyle{F(1)-F(0)}\\
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| − | &&\\
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| − | & = & \displaystyle{e^{\arctan(1)}-e^{\arctan(0)}}\\
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| − | &&\\
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| − | & = & \displaystyle{e^{\frac{\pi}{4}}-e^0}\\
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| − | &&\\
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| − | & = & \displaystyle{e^{\frac{\pi}{4}}-1.}
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| − | \end{array}</math>
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| − | |}
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| − | '''(c)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |Using the Fundamental Theorem of Calculus Part 1 and the Chain Rule, we have
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| − | | <math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\frac{d}{dx}\bigg(\frac{1}{x}\bigg).</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | |Hence, we have
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| − | | <math>\frac{d}{dx}\int_1^{\frac{1}{x}} \sin t~dt=\sin\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg).</math>
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| − | |}
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)''' See above
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| − | | '''(b)''' <math>e^{\frac{\pi}{4}}-1</math>
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| − | | '''(c)''' <math>\sin\bigg(\frac{1}{x}\bigg)\bigg(-\frac{1}{x^2}\bigg)</math>
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| − | |}
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| | [[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] | | [[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']] |
