022 Exam 2 Sample A, Problem 2 - Revision history

MathAdmin at 14:49, 16 May 2015

2015-05-16T14:49:52Z

MathAdmin at 22:45, 15 May 2015

2015-05-15T22:45:31Z

MathAdmin at 23:11, 14 May 2015

2015-05-14T23:11:53Z

MathAdmin: Created page with " Find the antiderivative of $y\,=\,3x^2-12x+8.$ {| class="mw-collapsible mw-collapsed" style = "text-align:..."

2015-05-14T23:11:32Z

Created page with " Find the antiderivative of <math style="vertical-align: -10%">y\,=\,3x^2-12x+8.</math> {| class="mw-collapsible mw-collapsed" style = "text-align:..."

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Find the antiderivative of  <math style="vertical-align: -10%">y\,=\,3x^2-12x+8.</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
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|We only require some fundamental rules for antiderivatives/integrals. We have the power rule:
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::<math style="vertical-align: -70%;">\int x^n\,dx\,=\,\frac{x^{n+1}}{n+1} +C,</math>  for <math style="vertical-align: -25%;">x\neq -1.</math>
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|Also, like derivatives, multiplication by a constant and addition/subtraction are respected by the antiderivative:
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|
::<math style="vertical-align: -70%;">\int c\cdot f(x)+g(x)\,dx\,=\,c\int f(x)\,dx+\int g(x)\,dx.</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Solution:  
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|We can apply the rules listed above to find
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|
::
<math>\begin{array}{rcl}
\int y\, dx & = & \int3x^{2}-12x+8\, dx\\
& = & 3\cdot \frac{x^3}{3}-12\cdot \frac{x^2}{2}+8x+C\\
& = & x^3-6x^2+8x+C.\end{array}</math>
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|Do '''not''' forget the constant when evaluating an antiderivative (i.e., an integral without upper and lower bounds)!

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|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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::<math>x^3-6x^2+8x+C.</math>
[[022_Exam_2_Sample_A|'''Return to Sample Exam''']]

@@ Line 37: / Line 37: @@
+|
 ::<math>x^3-6x^2+8x+C.</math>
 [[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

@@ Line 7: / Line 7: @@
 |-
+|
-::<math style="vertical-align: -70%;">\int x^n\,dx\,=\,\frac{x^{n+1}}{n+1} +C,</math>&thinsp; for <math style="vertical-align: -25%;">x\neq -1.</math>
+::<math style="vertical-align: -70%;">\int x^n\,dx\,=\,\frac{x^{n+1}}{n+1} +C,</math>&thinsp; for <math style="vertical-align: -25%;">n\neq -1.</math>
 |-
 |Also, like derivatives, multiplication by a constant and addition/subtraction are respected by the antiderivative:

@@ Line 24: / Line 24: @@
 <math>\begin{array}{rcl}
 \int y\, dx & = & \int3x^{2}-12x+8\, dx\\
   & = & 3\cdot \frac{x^3}{3}-12\cdot \frac{x^2}{2}+8x+C\\
- & = & x^3-6x^2+8x+C.\end{array}</math>
+\\
 |-
 |Do <u>'''not'''</u> forget the constant when evaluating an antiderivative (i.e., an integral without upper and lower bounds)!

022 Exam 2 Sample A, Problem 2 - Revision history

MathAdmin at 14:49, 16 May 2015

MathAdmin at 22:45, 15 May 2015

MathAdmin at 23:11, 14 May 2015

MathAdmin: Created page with " Find the antiderivative of y\,=\,3x^2-12x+8. {| class="mw-collapsible mw-collapsed" style = "text-align:..."

MathAdmin: Created page with " Find the antiderivative of $y\,=\,3x^2-12x+8.$ {| class="mw-collapsible mw-collapsed" style = "text-align:..."