009A Sample Final A, Problem 5 - Revision history

MathAdmin at 20:24, 6 May 2015

2015-05-06T20:24:04Z

MathAdmin at 04:08, 13 April 2015

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MathAdmin at 16:48, 2 April 2015

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MathAdmin at 19:46, 26 March 2015

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MathAdmin at 19:45, 26 March 2015

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MathAdmin at 19:07, 26 March 2015

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MathAdmin at 19:02, 26 March 2015

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MathAdmin at 18:50, 26 March 2015

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MathAdmin at 18:50, 26 March 2015

2015-03-26T18:50:03Z

MathAdmin at 18:49, 26 March 2015

2015-03-26T18:49:04Z

@@ Line 38: / Line 38: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Find the Derivatives and Their Roots: &nbsp;
+!Step 1: &nbsp;
 |-
-|Note that
+|'''Find the Derivatives and Their Roots.'''  Note that
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>f'(x)=x^{2}-4x-5 = (x-5)(x+1),</math>
@@ Line 52: / Line 52: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Produce Sign Charts and Evaluate: &nbsp;
 |-
-|Since all of our tests rely on the signs of our derivatives, we need to produce sign charts.  For the first derivative, we can test values below -1, between -1 and 5 and above 5.  For example:
+|'''Produce Sign Charts and Evaluate.''' Since all of our tests rely on the signs of our derivatives, we need to produce sign charts.  For the first derivative, we can test values below -1, between -1 and 5 and above 5.  For example:
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>f'(-10)=(-)(-)=(+),\quad f'(0)=(-)(+)=(-), \quad f'(10)=(+)(+)=(+).</math>
@@ Line 118: / Line 119: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Graph: &nbsp;
+!Step 3: &nbsp;
 |-
-|This is part (e) of the problem. We wish to use all our results.  In the image, the dots represent the two local extrema at ''x'' = -1 and ''x'' = 5, as well as the inflection point at ''x'' = 2.  The graph is drawn in blue where it is concave downward, and in red where it is concave upward.
+|'''Graph.'''This is part (e) of the problem. We wish to use all our results.  In the image, the dots represent the two local extrema at ''x'' = -1 and ''x'' = 5, as well as the inflection point at ''x'' = 2.  The graph is drawn in blue where it is concave downward, and in red where it is concave upward.
 |-
 |<br>[[File:SampFinA_5GP.png|center|500px]]

@@ Line 18: / Line 18: @@
 ! Foundations: &nbsp;
 |-
-|We learn a lot about the shape of a function's graph from its derivatives.  When a first derivative is positive, the function is increasing (heading uphill).  When the first derivative is negative, it is decreasing (heading downhill).  Of particular interest is when the first derivative at a point is zero.   If ''f'' '(''z'') = 0 a point ''z'', and the first derivative splits around it (either ''f'' '(x) < 0 for ''x'' < ''z'' and ''f'' '(x) > 0 for ''x'' > ''z'' or   ''f'' '(x) > 0 for ''x'' < ''z'' and ''f'' '(x) < 0 for ''x'' > ''z''), then the point (''z'',''f''(''z'')) is a '''local maximum''' or '''minimum''', respectively.
+|We learn a lot about the shape of a function's graph from its derivatives.  When a first derivative is positive, the function is increasing (heading uphill).  When the first derivative is negative, it is decreasing (heading downhill).  Of particular interest is when the first derivative at a point is zero.   If ''f'' '(''z'') = 0 at a point ''z'', and the first derivative splits around it (either ''f'' '(x) < 0 for ''x'' < ''z'' and ''f'' '(x) > 0 for ''x'' > ''z'' or   ''f'' '(x) > 0 for ''x'' < ''z'' and ''f'' '(x) < 0 for ''x'' > ''z''), then the point (''z'',''f''(''z'')) is a '''local maximum''' or '''minimum''', respectively.
 |-
 |The second derivative tells us how the ''first derivative'' is changing.  If the second derivative is positive, the first derivative (the slope of the tangent line) is increasing.  This is equivalent to the graph "turning left" if we consider moving from negative ''x''-values to positive.  We call this "concave up". The parabola ''y'' = ''x''<span style="font-size:85%"><sup>2</sup></span> is an example of a purely concave up graph, and its second derivative is the constant function ''y'' " = 2.

@@ Line 81: / Line 81: @@
 |This gives us the following answers:
 |-
-|(a) The function is increasing on <math style="vertical-align: -22%;">-(\infty,-1) </math> and <math style="vertical-align: -22%;">(5,\infty)</math>, and decreasing on <math  style="vertical-align: -22%;">(-1,5)</math>.
+|(a) The function is increasing on <math style="vertical-align: -22%;">(\infty,-1) </math> and <math style="vertical-align: -22%;">(5,\infty)</math>, and decreasing on <math  style="vertical-align: -22%;">(-1,5)</math>.
 |-
 |(b) The first derivative test shows

@@ Line 118: / Line 118: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Graph (Part (e)): &nbsp;
+!Graph: &nbsp;
 |-
-|We wish to use all our results.  In the image, the dots represent the two local extrema at ''x'' = -1 and ''x'' = 5, as well as the inflection point at ''x'' = 2.  The graph is drawn in blue where it is concave downward, and in red where it is concave upward.
+|This is part (e) of the problem. We wish to use all our results.  In the image, the dots represent the two local extrema at ''x'' = -1 and ''x'' = 5, as well as the inflection point at ''x'' = 2.  The graph is drawn in blue where it is concave downward, and in red where it is concave upward.
 |-
 |<br>[[File:SampFinA_5GP.png|center|500px]]

@@ Line 38: / Line 38: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Find the Derivatives and Roots: &nbsp;
+!Find the Derivatives and Their Roots: &nbsp;
 |-
 |Note that

@@ Line 116: / Line 116: @@
 |(d) The function has an inflection point at <math  style="vertical-align: -67%;">(2,f(2))=\left(2,-\frac{11}{3}\right).</math>
 |}
 [[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']]

@@ Line 112: / Line 112: @@
 |This gives us the following answers:
 |-
-|(c) The function is concave downward on <math style="vertical-align: -22%;">(-\infty,2)</math> and concave upward on <math style="vertical-align: -24%;">(2,\infty)</math>.
+|(c) The function is concave downward on <math style="vertical-align: -22%;">(-\infty,2)</math> and concave upward on <math style="vertical-align: -22%;">(2,\infty)</math>.
 |-
-|(d) The function has an inflection point at <math  style="vertical-align: -69%;">(2,f(2))=\left(2,-\frac{11}{3}\right).</math>
+|(d) The function has an inflection point at <math  style="vertical-align: -67%;">(2,f(2))=\left(2,-\frac{11}{3}\right).</math>
 |}
 [[009A_Sample_Final_A|'''<u>Return to Sample Exam</u>''']]

@@ Line 89: / Line 89: @@
 |is a local maximum, while
 |-
-|&nbsp;&nbsp;&nbsp;&nbsp; <math>f(5)=\frac{125}{3}-50-25+\frac{35}{3}=-75+\frac{90}{3}=-45</math>
+|&nbsp;&nbsp;&nbsp;&nbsp; <math>f(5)=\frac{125}{3}-50-25+\frac{35}{\,3}=-75+\frac{160}{3}=-\,\frac{65}{\,3}.</math>
 |-
 |is a local minimum.