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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;"
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| − | !Foundations:
| + | [[009B Sample Final 2, Problem 1 Solution|'''<u>Solution</u>''']] |
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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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| − | !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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| − | !Step 1:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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| − | '''(c)'''
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| − | !Step 1:
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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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)'''
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| − | |'''(c)'''
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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>''']] |
