MathAdmin at 01:13, 7 June 2015

2015-06-07T01:13:07Z

MathAdmin: Created page with " Sketch the curve, including all relative extrema and points of inflection: $y = 3x^4 - 4x^3$ . {| class="mw-collaps..."

2015-06-06T17:51:15Z

Created page with " Sketch the curve, including all relative extrema and points of inflection: <math style="vertical-align: -4px">y = 3x^4 - 4x^3</math>. {| class="mw-collaps..."

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Sketch the curve, including all relative extrema and points of inflection: <math style="vertical-align: -4px">y = 3x^4 - 4x^3</math>.

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations:  
|-
|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''2 is an example of a purely concave up graph, and its second derivative is the constant function ''y'' " = 2.
|-
|If the second derivative is negative, then the first derivative is decreasing. This means we are turning right as we move from negative ''x''-values to positive. This is called "concave down". The inverted parabola ''y'' = -''x''2 is an example of a purely concave down graph.
|-
|A point ''z'' where the second derivative is zero, and the sign of the second 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 an inflection point.
|-
| Of course, there are tests we use to find local extrema (maxima and minima, which is the plural of maximum and minimum). We are assuming the function ''f'' is continuous and differentiable in an interval containing the point ''x''0.
|-
|'''First Derivative Test:''' If at a point ''x''0, ''f'' '(''x''0) = 0, and ''f'' '(''x'') < 0 for ''x'' < ''x''0 while ''f'' '(''x'') > 0 for ''x'' > ''x''0, then ''f''(''x''0) is a local minimum.
|-
|On the other hand, if ''f'' '(''x''0) = 0, and ''f'' '(''x'') > 0 for ''x'' < ''x''0 while ''f'' '(''x'') < 0 for ''x'' > ''x''0, then ''f''(''x''0) is a local maximum.
|-
|'''Second Derivative Test:''' If at a point ''x''0, ''f'' '(''x''0) = 0, and ''f'' "(''x''0) > 0, then ''f''(''x''0) is a local minimum. On the other hand, if ''f'' "(''x''0) < 0, then ''f''(''x''0) is a local maximum. If ''f'' "(''x''0) = 0, the test is inconclusive.
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|'''Find the Derivatives and Their Roots.''' Note that
|-
|
::<math>y'\,=\,12x^3-12x^2\,=\,12x^2(x-1).</math>
|-
|This has roots <math style="vertical-align: 0px">0</math> and <math style="vertical-align: 0px">1</math>.
|-
|On the other hand,
|-
|
::<math>y''\,=,36x^2-24x\,=\,12x(3x-2).</math>
|-
|This has roots <math style="vertical-align: 0px">0</math> and <math style="vertical-align: -5px">2/3</math>.
|-
|For graphing, I would also note that <math style="vertical-align: -4px">y=x^3(3x-4)</math>, which has roots <math style="vertical-align: 0px">0</math> and <math style="vertical-align: -5px">4/3</math>.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

!Step 2:  
|-
|'''Produce Sign Charts and Evaluate.''' It should be clear that the function we're graphing is "roughly" like <math style="vertical-align: 0px">x^4</math>, which should tell you it goes to positive infinity off to the left and right.
 
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 <math style="vertical-align: 0px">0</math>, between <math style="vertical-align: 0px">0</math> and <math style="vertical-align: -1px">1</math> and above <math style="vertical-align: -1px">1</math>. Using the factored version of each derivative, we can evaluate quickly. For example:
|-
|     <math>f'(-10)=(+)(-)=(-),\quad f'(1/2)=(+)(-)=(-), \quad f'(10)=(+)(+)=(+).</math>
|-
|From this, we can build a sign chart:
|-
|<table border="1" cellspacing="0" cellpadding="6" align = "center">
<tr>
<td align = "center"><math> x:</math></td>
<td align = "center"><math> x<0 </math></td>
<td align = "center"><math> x=0 </math></td>
<td align = "center"><math> 0<x<1 </math></td>
<td align = "center"><math> x=1 </math></td>
<td align = "center"><math>x>1</math></td>
</tr>
<tr>
<td align = "center"><math> y':</math></td>
<td align = "center"><math> (-) </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> (-) </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> (+)</math></td>
</tr>
</table>
|-
|This tells us the point at the origin is '''not''' a local extrema, as the first derivative does not '''split''' around it. On the other hand, by the first derivative test we have a local minimum at <math style="vertical-align: -5px">(1,-1)</math>.
|-
|The process is similar for the second derivative. Here, we can test values below <math style="vertical-align: 0px">0</math>, between <math style="vertical-align: 0px">0</math> and <math style="vertical-align: -5px">2/3</math> and above <math style="vertical-align: -5px">2/3</math>. Using the factored version of each derivative, we evaluate test points:
|-
|     <math>f'(-10)=(-)(-)=(+),\quad f'(1/2)=(+)(-)=(-), \quad f'(10)=(+)(+)=(+).</math>
|-
|From this, we can build a sign chart:
|-
|<table border="1" cellspacing="0" cellpadding="6" align = "center">
<tr>
<td align = "center"><math> x:</math></td>
<td align = "center"><math> x<0 </math></td>
<td align = "center"><math> x=0 </math></td>
<td align = "center"><math> 0<x<2/3 </math></td>
<td align = "center"><math> x=2/3 </math></td>
<td align = "center"><math>x>2/3</math></td>
</tr>
<tr>
<td align = "center"><math> y'':</math></td>
<td align = "center"><math> (+) </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> (-) </math></td>
<td align = "center"><math> 0 </math></td>
<td align = "center"><math> (+)</math></td>
</tr>
</table>
|-
|This tells us the function is concave downward when <math style="vertical-align: -5px"> 0<x<2/3 </math>, and concave upwards everywhere else. Additionally, there is an inflection point at <math style="vertical-align: -15px">\left(\frac{2}{3},\frac{16}{81}\right)</math>, although a teacher may not require you to compute the actual <math style="vertical-align: -4px">y</math>-value.
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 3:  

|-
|'''Graph:''' The graph is drawn in blue where it is concave downward, and in red where it is concave upward. You should probably label the local minimum and inflection point, if you have time.
|-
| [[File:022_3_A_6.png|center|400px]]
|-
|}

[[022_Sample_Final_A|'''Return to Sample Exam''']]

@@ Line 31: / Line 31: @@
 ::<math>y'\,=\,12x^3-12x^2\,=\,12x^2(x-1).</math>
 |-
-|This has roots <math style="vertical-align: 0px">0</math> and <math style="vertical-align: 0px">1</math>.
+|This has roots <math style="vertical-align: 0px">0</math> and <math style="vertical-align: -1px">1</math>.
 |-
 |On the other hand,

022 Sample Final A, Problem 6 - Revision history

MathAdmin at 01:13, 7 June 2015

MathAdmin: Created page with " Sketch the curve, including all relative extrema and points of inflection: y = 3x^4 - 4x^3. {| class="mw-collaps..."

MathAdmin: Created page with " Sketch the curve, including all relative extrema and points of inflection: $y = 3x^4 - 4x^3$ . {| class="mw-collaps..."