022 Exam 1 Sample A, Problem 4 - Revision history

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← Older revision		Revision as of 05:03, 13 April 2015
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−	Determine the intervals where the function  <math style="vertical-align: -16%">h(x)=2x^{4}-x^{2}</math>	+	Determine the intervals where the function  <math style="vertical-align: -20%">h(x)=2x^{4}-x^{2}</math>
	is increasing or decreasing.		is increasing or decreasing.

← Older revision		Revision as of 05:02, 13 April 2015
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−	~~<span class="exam">[[022_Exam_1_Sample_A,_Problem_4 \|<span class="biglink"> Problem 4. </span>]]~~ Determine the intervals where the function  <math style="vertical-align: -16%">h(x)=2x^{4}-x^{2}</math>	+	Determine the intervals where the function  <math style="vertical-align: -16%">h(x)=2x^{4}-x^{2}</math>
	is increasing or decreasing.		is increasing or decreasing.

@@ Line 8: / Line 8: @@
 <br>
 |-
-|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -25%">g'(0)=0</math>, but is strictly positive every else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math>&thinsp; is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
+|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -25%">g'(0)=0</math>, but is strictly positive everywhere else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math>&thinsp; is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
 |}

@@ Line 8: / Line 8: @@
 <br>
 |-
-|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -30%">g'(0)=0</math>, but is strictly positive every else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math>&thinsp; is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
+|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -25%">g'(0)=0</math>, but is strictly positive every else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math>&thinsp; is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
 |}

@@ Line 5: / Line 5: @@
 ! Foundations: &nbsp;
 |-
-|When a first derivative is positive, the function is increasing (heading uphill).  When the first derivative is negative, it is decreasing (heading downhill).  When the first derivative is <math style="vertical-align: -20%">0,</math> it is not quite so clear.   If <math style="vertical-align: -22%">f'(z)=0</math>&thinsp; at a point <math style="vertical-align: 0%">z</math>, and the first derivative splits around it (either <math style="vertical-align: -23%">f'(x)<0</math>&thinsp; for <math style="vertical-align: 0%">x<z</math> and <math style="vertical-align: -23%">f'(x)>0</math>&thinsp; for <math style="vertical-align: 0%">x > z</math>, or <math style="vertical-align: -23%">f'(x) > 0</math>&thinsp; for <math style="vertical-align: 0%">x< z</math> and <math style="vertical-align: -23%">f'(x) < 0</math>&thinsp; for <math style="vertical-align: 0%">x> z</math>), then the point <math style="vertical-align: -20%">(z,f(z))</math> is a '''local maximum''' or '''minimum''', respectively, and is neither increasing or decreasing at that point.
+|When a first derivative is positive, the function is increasing (heading uphill).  When the first derivative is negative, it is decreasing (heading downhill).  When the first derivative is <math style="vertical-align: -18%">0,</math> it is not quite so clear.   If <math style="vertical-align: -22%">f'(z)=0</math>&thinsp; at a point <math style="vertical-align: 0%">z</math>, and the first derivative splits around it (either <math style="vertical-align: -22%">f'(x)<0</math>&thinsp; for <math style="vertical-align: 0%">x<z</math> and <math style="vertical-align: -22%">f'(x)>0</math>&thinsp; for <math style="vertical-align: 0%">x > z</math>, or <math style="vertical-align: -22%">f'(x) > 0</math>&thinsp; for <math style="vertical-align: 0%">x< z</math> and <math style="vertical-align: -22%">f'(x) < 0</math>&thinsp; for <math style="vertical-align: 0%">x> z</math>), then the point <math style="vertical-align: -20%">(z,f(z))</math> is a '''local maximum''' or '''minimum''', respectively, and is neither increasing or decreasing at that point.
 <br>
 |-

@@ Line 8: / Line 8: @@
 <br>
 |-
-|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -30%">g'(0)=0</math>, but is strictly positive every else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math> is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
+|On the other hand, if the first derivative does not split around <math style="vertical-align: 0%">z</math>, then it will be increasing or decreasing at that point based on the derivative of the adjacent intervals.  For example, <math style="vertical-align: -25%">g(x)=x^3</math> has the derivative <math style="vertical-align: -20%">g'(x)=3x^2</math>.  Thus, <math style="vertical-align: -30%">g'(0)=0</math>, but is strictly positive every else.  As a result, <math style="vertical-align: -20%">g(x)=x^3</math>&thinsp; is increasing on <math style="vertical-align: -20%">(-\infty,\infty)</math>.
 |}
@@ Line 14: / Line 14: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Find the Derivatives and Their Roots: &nbsp;
+!Find the Roots of the First Derivative: &nbsp;
 |-
 |Note that
 |}
 [[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]