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| | <span class="exam">(d) Sketch the shape of the graph of <math style="vertical-align: -4px">f.</math> | | <span class="exam">(d) Sketch the shape of the graph of <math style="vertical-align: -4px">f.</math> |
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| | + | [[009A Sample Final 3, Problem 6 Solution|'''<u>Solution</u>''']] |
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Foundations:
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| − | |'''1.''' <math style="vertical-align: -5px">f(x)</math> is increasing when <math style="vertical-align: -5px">f'(x)>0</math> and <math style="vertical-align: -5px">f(x)</math> is decreasing when <math style="vertical-align: -5px">f'(x)<0.</math>
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| − | |'''2. The First Derivative Test''' tells us when we have a local maximum or local minimum.
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| − | |'''3.''' <math style="vertical-align: -5px">f(x)</math> is concave up when <math style="vertical-align: -5px">f''(x)>0</math> and <math style="vertical-align: -5px">f(x)</math> is concave down when <math style="vertical-align: -5px">f''(x)<0.</math>
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| | + | [[009A Sample Final 3, Problem 6 Detailed Solution|'''<u>Detailed Solution</u>''']] |
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| − | '''Solution:'''
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| − | '''(a)'''
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 1:
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| − | |We start by taking the derivative of <math style="vertical-align: -5px">f(x).</math> We have <math style="vertical-align: -5px">f'(x)=24x^2-4x^3.</math>
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| − | |Now, we set <math style="vertical-align: -5px">f'(x)=0.</math> So, we have <math style="vertical-align: -6px">0=4x^2(6-x).</math>
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| − | |Hence, we have <math style="vertical-align: 0px">x=0</math> and <math style="vertical-align: -1px">x=6.</math>
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| − | |So, these values of <math style="vertical-align: 0px">x</math> break up the number line into 3 intervals: <math style="vertical-align: -5px">(-\infty,0),(0,6),(6,\infty).</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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| − | |To check whether the function is increasing or decreasing in these intervals, we use testpoints.
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| − | |For <math style="vertical-align: -5px">x=-1,~f'(x)=28>0.</math>
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| − | |For <math style="vertical-align: -5px">x=1,~f'(x)=20>0.</math>
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| − | |For <math style="vertical-align: -5px">x=7,~f'(x)=-196<0.</math>
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| − | |Thus, <math style="vertical-align: -5px">f(x)</math> is increasing on <math style="vertical-align: -5px">(-\infty,6)</math> and decreasing on <math style="vertical-align: -5px">(6,\infty).</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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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Step 2:
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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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| − | {| 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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| − | !(d):
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| − | |Insert graph
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| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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| − | !Final Answer:
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| − | | '''(a)''' <math style="vertical-align: -5px">f(x)</math> is increasing on <math style="vertical-align: -5px">(-\infty,6)</math> and decreasing on <math style="vertical-align: -5px">(6,\infty).</math>
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| − | | '''(b)'''
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| − | | '''(c)'''
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| − | | '''(d)''' See above
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| − | |}
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| | [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] | | [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']] |
Let
(a) Over what 
-intervals is 
increasing/decreasing?
(b) Find all critical points 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}
and test each for local maximum and local minimum.
(c) Over what 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 x}
-intervals is 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}
concave up/down?
(d) Sketch the shape of the graph 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.}
Solution
Detailed Solution
Return to Sample Exam