Difference between revisions of "009A Sample Final 1, Problem 9"

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!Step 1:    
 
!Step 1:    
 
|-
 
|-
−
|We start by taking the derivative of <math>f(x)</math>. We have <math>f'(x)=3x^2-12x</math>.
+
|We start by taking the derivative of <math style="vertical-align: -5px">f(x)</math>. We have <math style="vertical-align: -5px">f'(x)=3x^2-12x</math>.
 
|-
 
|-
−
|Now, we set <math>f'(x)=0</math>. So, we have <math>0=3x(x-4)</math>.
+
|Now, we set <math style="vertical-align: -5px">f'(x)=0</math>. So, we have <math style="vertical-align: -5px">0=3x(x-4)</math>.
 
|-
 
|-
−
|Hence, we have <math>x=0</math> and <math>x=4</math>.
+
|Hence, we have <math style="vertical-align: -1px">x=0</math> and <math style="vertical-align: -1px">x=4</math>.
 
|-
 
|-
−
|So, these values of <math>x</math> break up the number line into 3 intervals: <math>(-\infty,0),(0,4),(4,\infty)</math>.  
+
|So, these values of <math style="vertical-align: -1px">x</math> break up the number line into 3 intervals: <math style="vertical-align: -3px">(-\infty,0),(0,4),(4,\infty)</math>.  
 
|}
 
|}
  
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|To check whether the function is increasing or decreasing in these intervals, we use testpoints.  
 
|To check whether the function is increasing or decreasing in these intervals, we use testpoints.  
 
|-
 
|-
−
|For <math>x=-1</math>, <math>f'(x)=15>0</math>.
+
|For <math style="vertical-align: -5px">x=-1,~f'(x)=15>0</math>.
 
|-
 
|-
−
|For <math>x=1</math>, <math>f'(x)=-9<0</math>.
+
|For <math style="vertical-align: -5px">x=1,~f'(x)=-9<0</math>.
 
|-
 
|-
−
|For <math>x=5</math>, <math>f'(x)=15>0</math>.
+
|For <math style="vertical-align: -5px">x=5,~f'(x)=15>0</math>.
 
|-
 
|-
−
|Thus, <math>f(x)</math> is increasing on <math>(-\infty,0),(4,\infty)</math> and decreasing on <math>(0,4)</math>.
+
|Thus, <math style="vertical-align: -5px">f(x)</math> is increasing on <math style="vertical-align: -5px">(-\infty,0),(4,\infty)</math> and decreasing on <math style="vertical-align: -5px">(0,4)</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|The local maximum occurs at <math>x=0</math> and the local minimum occurs at <math>x=4</math>.
+
|The local maximum occurs at <math style="vertical-align: -1px">x=0</math> and the local minimum occurs at <math style="vertical-align: -1px">x=4</math>.
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|So, the local maximum value is <math>f(0)=5</math> and the local minimum value is <math>f(4)=-27</math>.
+
|So, the local maximum value is <math style="vertical-align: -5px">f(0)=5</math> and the local minimum value is <math style="vertical-align: -5px">f(4)=-27</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|To find the intervals when the function is concave up or concave down, we need to find <math>f''(x)</math>.
+
|To find the intervals when the function is concave up or concave down, we need to find <math style="vertical-align: -5px">f''(x)</math>.
 
|-
 
|-
−
|We have <math>f''(x)=6x-12</math>.
+
|We have <math style="vertical-align: -5px">f''(x)=6x-12</math>.
 
|-
 
|-
−
|We set <math>f''(x)=0</math>.
+
|We set <math style="vertical-align: -5px">f''(x)=0</math>.
 
|-
 
|-
−
|So, we have <math>0=6x-12</math>. Hence, <math>x=2</math>.
+
|So, we have <math style="vertical-align: -1px">0=6x-12</math>. Hence, <math style="vertical-align: -1px">x=2</math>.
 
|-
 
|-
−
|This value breaks up the number line into two intervals: <math>(-\infty,2),(2,\infty)</math>.
+
|This value breaks up the number line into two intervals: <math style="vertical-align: -5px">(-\infty,2),(2,\infty)</math>.
 
|}
 
|}
  
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|Again, we use test points in these two intervals.
 
|Again, we use test points in these two intervals.
 
|-
 
|-
−
|For <math>x=0</math>, we have <math>f''(x)=-12<0</math>.
+
|For <math style="vertical-align: -1px">x=0</math>, we have <math style="vertical-align: -5px">f''(x)=-12<0</math>.
 
|-
 
|-
−
|For <math>x=3</math>, we have <math>f''(x)=6>0</math>.
+
|For <math style="vertical-align: -1px">x=3</math>, we have <math style="vertical-align: -5px">f''(x)=6>0</math>.
 
|-
 
|-
−
|Thus, <math>f(x)</math> is concave up on the interval <math>(2,\infty)</math> and concave down on the interval <math>(-\infty,2)</math>.
+
|Thus, <math style="vertical-align: -5px">f(x)</math> is concave up on the interval <math style="vertical-align: -5px">(2,\infty)</math> and concave down on the interval <math style="vertical-align: -5px">(-\infty,2)</math>.
 
|}
 
|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
| Using the information from part '''(c)''', there is one inflection point that occurs at <math>x=2</math>.
+
| Using the information from part '''(c)''', there is one inflection point that occurs at <math style="vertical-align: -1px">x=2</math>.
 
|-
 
|-
−
|Now, we have <math>f(2)=8-24+5=-11</math>.
+
|Now, we have <math style="vertical-align: -5px">f(2)=8-24+5=-11</math>.
 
|-
 
|-
−
|So, the inflection point is <math>(2,-11)</math>.
+
|So, the inflection point is <math style="vertical-align: -5px">(2,-11)</math>.
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|'''(a)''' <math>f(x)</math> is increasing on <math>(-\infty,0),(4,\infty)</math> and decreasing on <math>(0,4)</math>.
+
|'''(a)''' <math style="vertical-align: -5px">f(x)</math> is increasing on <math style="vertical-align: -5px">(-\infty,0),(4,\infty)</math> and decreasing on <math style="vertical-align: -5px">(0,4)</math>.
 
|-
 
|-
−
|'''(b)''' The local maximum value is <math>f(0)=5</math> and the local minimum value is <math>f(4)=-27</math>.  
+
|'''(b)''' The local maximum value is <math style="vertical-align: -5px">f(0)=5</math> and the local minimum value is <math style="vertical-align: -5px">f(4)=-27</math>.
 
|-
 
|-
−
|'''(c)''' <math>f(x)</math> is concave up on the interval <math>(2,\infty)</math> and concave down on the interval <math>(-\infty,2)</math>.
+
|'''(c)''' <math style="vertical-align: -5px">f(x)</math> is concave up on the interval <math style="vertical-align: -5px">(2,\infty)</math> and concave down on the interval <math style="vertical-align: -5px">(-\infty,2)</math>.
 
|-
 
|-
−
|'''(d)''' <math>(2,-11)</math>
+
|'''(d)''' <math style="vertical-align: -5px">(2,-11)</math>
 
|-
 
|-
−
|'''(e)''' See part '''(e)''' above for graph.
+
|'''(e)''' See Step 1 for graph.
 
|}
 
|}
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:56, 22 February 2016

Given the function ,

a) Find the intervals in which the function increases or decreases.

b) Find the local maximum and local minimum values.

c) Find the intervals in which the function concaves upward or concaves downward.

d) Find the inflection point(s).

e) Use the above information (a) to (d) to sketch the graph of .

Foundations:  

Solution:

(a)

Step 1:  
We start by taking the derivative of . We have .
Now, we set . So, we have 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 0=3x(x-4)} .
Hence, we have 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=0} 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 x=4} .
So, these values 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 x} break up the number line into 3 intervals: 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 (-\infty,0),(0,4),(4,\infty)} .
Step 2:  
To check whether the function is increasing or decreasing in these intervals, we use testpoints.
For 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=-1,~f'(x)=15>0} .
For 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=1,~f'(x)=-9<0} .
For 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=5,~f'(x)=15>0} .
Thus, 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)} is increasing 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 (-\infty,0),(4,\infty)} and decreasing 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 (0,4)} .

(b)

Step 1:  
The local maximum occurs at 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=0} and the local minimum occurs at 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=4} .
Step 2:  
So, the local maximum value 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(0)=5} and the local minimum value 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(4)=-27} .

(c)

Step 1:  
To find the intervals when the function is concave up or concave down, we need to find 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)} .
We have 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)=6x-12} .
We set 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)=0} .
So, we have 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 0=6x-12} . Hence, 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=2} .
This value breaks up the number line into two intervals: 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 (-\infty,2),(2,\infty)} .
Step 2:  
Again, we use test points in these two intervals.
For 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=0} , we have 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)=-12<0} .
For 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=3} , we have 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)=6>0} .
Thus, 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)} is concave up on the interval 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 (2,\infty)} and concave down on the interval 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 (-\infty,2)} .

(d)

Step 1:  
Using the information from part (c), there is one inflection point that occurs at 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=2} .
Now, we have 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(2)=8-24+5=-11} .
So, the inflection point 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 (2,-11)} .

(e)

Step 1:  
Insert sketch here.
Final Answer:  
(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)} is increasing 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 (-\infty,0),(4,\infty)} and decreasing 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 (0,4)} .
(b) The local maximum value 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(0)=5} and the local minimum value 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(4)=-27} .
(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 f(x)} is concave up on the interval 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 (2,\infty)} and concave down on the interval 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 (-\infty,2)} .
(d) 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 (2,-11)}
(e) See Step 1 for graph.

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