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

Revision as of 11:23, 18 April 2016

Given the function 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)=x^3-6x^2+5} ,

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 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 y=f(x)} .

Foundations:
Recall:
1. 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 when 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} 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)} is decreasing when 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.}
2. The First Derivative Test tells us when we have a local maximum or local minimum.
3. 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 when 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} 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)} is concave down when 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.}
4. Inflection points occur when 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.}

Solution:

(a)

Step 1:
We start by taking the derivative 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(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)=3x^2-12x.}
Now, 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=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)\cup (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:

By the First Derivative Test, 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)
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 $(2,-11).$

(e)

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 graph in (e).

Return to Sample Exam

@@ Line 1: / Line 1: @@
 <span class="exam">Given the function <math style="vertical-align: -5px">f(x)=x^3-6x^2+5</math>,
-<span class="exam">a) Find the intervals in which the function increases or decreases.
+::<span class="exam">a) Find the intervals in which the function increases or decreases.
-<span class="exam">b) Find the local maximum and local minimum values.
+::<span class="exam">b) Find the local maximum and local minimum values.
-<span class="exam">c) Find the intervals in which the function concaves upward or concaves downward.
+::<span class="exam">c) Find the intervals in which the function concaves upward or concaves downward.
-<span class="exam">d) Find the inflection point(s).
+::<span class="exam">d) Find the inflection point(s).
-<span class="exam">e) Use the above information (a) to (d) to sketch the graph of <math style="vertical-align: -5px">y=f(x)</math>.
+::<span class="exam">e) Use the above information (a) to (d) to sketch the graph of <math style="vertical-align: -5px">y=f(x)</math>.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 16: / Line 16: @@
 |Recall:
 |-
-|'''1.''' <math style="vertical-align: -5px">f(x)</math>&thinsp; is increasing when <math style="vertical-align: -5px">f'(x)>0</math>&thinsp; and <math style="vertical-align: -5px">f(x)</math>&thinsp; is decreasing when <math style="vertical-align: -5px">f'(x)<0.</math>
+|
+::'''1.''' <math style="vertical-align: -5px">f(x)</math>&thinsp; is increasing when <math style="vertical-align: -5px">f'(x)>0</math>&thinsp; and <math style="vertical-align: -5px">f(x)</math>&thinsp; is decreasing when <math style="vertical-align: -5px">f'(x)<0.</math>
 |-
-|'''2. The First Derivative Test''' tells us when we have a local maximum or local minimum.
+|
+::'''2. The First Derivative Test''' tells us when we have a local maximum or local minimum.
 |-
-|'''3.''' <math style="vertical-align: -5px">f(x)</math>&thinsp; is concave up when <math style="vertical-align: -5px">f''(x)>0</math>&thinsp; and <math style="vertical-align: -5px">f(x)</math>&thinsp; is concave down when <math style="vertical-align: -5px">f''(x)<0.</math>
+|
+::'''3.''' <math style="vertical-align: -5px">f(x)</math>&thinsp; is concave up when <math style="vertical-align: -5px">f''(x)>0</math>&thinsp; and <math style="vertical-align: -5px">f(x)</math>&thinsp; is concave down when <math style="vertical-align: -5px">f''(x)<0.</math>
 |-
-|'''4.''' Inflection points occur when <math style="vertical-align: -5px">f''(x)=0.</math>
+|
+::'''4.''' Inflection points occur when <math style="vertical-align: -5px">f''(x)=0.</math>
 |}
@@ Line 32: / Line 36: @@
 !Step 1: &nbsp;
 |-
-|We start by taking the derivative of <math style="vertical-align: -5px">f(x).</math>&thinsp; We have <math style="vertical-align: -5px">f'(x)=3x^2-12x.</math>
+|We start by taking the derivative of <math style="vertical-align: -5px">f(x).</math>&thinsp; We have
 |-
-|Now, we set <math style="vertical-align: -5px">f'(x)=0.</math>&thinsp; So, we have &thinsp;<math style="vertical-align: -3px">(-\infty,0),(0,4),(4,\infty).</math><math style="vertical-align: -6px">0=3x(x-4).</math>
+|
+::<math style="vertical-align: -5px">f'(x)=3x^2-12x.</math>
+|-
+|Now, we set <math style="vertical-align: -5px">f'(x)=0.</math>&thinsp; So, we have
+|-
+|
+::<math style="vertical-align: -6px">0=3x(x-4).</math>
 |-
 |Hence, we have <math style="vertical-align: 0px">x=0</math>&thinsp; and <math style="vertical-align: -1px">x=4.</math>
@@ Line 46: / Line 56: @@
 |To check whether the function is increasing or decreasing in these intervals, we use testpoints.
 |-
-|For <math style="vertical-align: -5px">x=-1,~f'(x)=15>0.</math>
+|
+::For <math style="vertical-align: -5px">x=-1,~f'(x)=15>0.</math>
 |-
-|For <math style="vertical-align: -5px">x=1,~f'(x)=-9<0.</math>
+|
+::For <math style="vertical-align: -5px">x=1,~f'(x)=-9<0.</math>
 |-
-|For <math style="vertical-align: -5px">x=5,~f'(x)=15>0.</math>
+|
+::For <math style="vertical-align: -5px">x=5,~f'(x)=15>0.</math>
 |-
 |Thus, <math style="vertical-align: -5px">f(x)</math>&thinsp; is increasing on <math style="vertical-align: -5px">(-\infty,0)\cup (4,\infty),</math>&thinsp; and decreasing on <math style="vertical-align: -5px">(0,4).</math>
@@ Line 76: / Line 89: @@
 |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 style="vertical-align: -5px">f''(x)=6x-12.</math>
+|We have
+|-
+|
+::<math style="vertical-align: -5px">f''(x)=6x-12.</math>
 |-
 |We set <math style="vertical-align: -5px">f''(x)=0.</math>
 |-
-|So, we have <math style="vertical-align: -1px">0=6x-12.</math> Hence, <math style="vertical-align: 0px">x=2.</math>
+|So, we have
+|-
+|
+::<math style="vertical-align: -1px">0=6x-12.</math>
 |-
-|This value breaks up the number line into two intervals: <math style="vertical-align: -5px">(-\infty,2),(2,\infty).</math>
+|Hence, <math style="vertical-align: 0px">x=2.</math> This value breaks up the number line into two intervals: <math style="vertical-align: -5px">(-\infty,2),(2,\infty).</math>
 |}
@@ Line 90: / Line 109: @@
 |Again, we use test points in these two intervals.
 |-
-|For <math style="vertical-align: -5px">x=0,</math>&thinsp; we have <math style="vertical-align: -5px">f''(x)=-12<0.</math>
+|
+::For <math style="vertical-align: -5px">x=0,</math>&thinsp; we have <math style="vertical-align: -5px">f''(x)=-12<0.</math>
 |-
-|For <math style="vertical-align: -5px">x=3,</math>&thinsp; we have <math style="vertical-align: -5px">f''(x)=6>0.</math>
+|
+::For <math style="vertical-align: -5px">x=3,</math>&thinsp; we have <math style="vertical-align: -5px">f''(x)=6>0.</math>
 |-
 |Thus, <math style="vertical-align: -5px">f(x)</math>&thinsp; 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>

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

Revision as of 11:23, 18 April 2016

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