009C Sample Final 2, Problem 6 - Revision history

Kayla Murray at 01:31, 3 December 2017

2017-12-03T01:31:02Z

2017-03-11T00:09:27Z

2017-03-05T18:12:19Z

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@@ Line 3: / Line 3: @@
 <span class="exam">(b) Express the definite integral &nbsp;<math style="vertical-align: -14px">\int_0^1 \sin(x^2)~dx</math>&nbsp; as a number series.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 2, Problem 6 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
+[[009C Sample Final 2, Problem 6 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 22: / Line 22: @@
 |-
 |So, the power series of &nbsp;<math style="vertical-align: -5px">\sin(x^2)</math> &nbsp; is
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 \displaystyle{\sin(x^2)} & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n(x^2)^{2n+1}}{(2n+1)!}}\\
@@ Line 28: / Line 30: @@
 & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}.}
 \end{array}</math>
 |}
@@ Line 35: / Line 39: @@
 |Now, to express the indefinite integral as a power series, we have
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 \displaystyle{\int \sin(x^2)~dx} & = & \displaystyle{\int \sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}~dx}\\
@@ Line 42: / Line 48: @@
 & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}.}
 \end{array}</math>
 |}
@@ Line 63: / Line 71: @@
 |Hence, we have
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 \displaystyle{\int_0^1 \sin(x^2)~dx} & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n (1)^{4n+3}}{(4n+3)(2n+1)!}-\sum_{n=0}^\infty \frac{(-1)^n (0)^{4n+3}}{(4n+3)(2n+1)!}}\\
@@ Line 70: / Line 80: @@
 & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n}{(4n+3)(2n+1)!}}
 \end{array}</math>
 |}

@@ Line 49: / Line 49: @@
 !Step 1: &nbsp;
 |-
+|From part (a), we have
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\int \sin(x^2)~dx=\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}.</math>
 |-
+|Now, we have
 |}
@@ Line 59: / Line 61: @@
 !Step 2: &nbsp;
 |-
+|Hence, we have
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 |}
@@ Line 70: / Line 78: @@
 |&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}</math>
 |-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math></math>
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\sum_{n=0}^\infty \frac{(-1)^n}{(4n+3)(2n+1)!}</math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 24: / Line 24: @@
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{} & = & \displaystyle{}\\
+\displaystyle{\sin(x^2)} & = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^n(x^2)^{2n+1}}{(2n+1)!}}\\
 &&\\
-& = & \displaystyle{}\\
+& = & \displaystyle{\sum_{n=0}^\infty \frac{(-1)^nx^{4n+2}}{(2n+1)!}.}
 \end{array}</math>
 |}
@@ Line 35: / Line 33: @@
 !Step 2: &nbsp;
 |-
+|Now, to express the indefinite integral as a power series, we have
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 |}
@@ Line 64: / Line 68: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\sum_{n=0}^\infty \frac{(-1)^n x^{4n+3}}{(4n+3)(2n+1)!}</math>
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math></math>
 |}
 [[009C_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 19: / Line 19: @@
 !Step 1: &nbsp;
 |-
+|The power series of &nbsp;<math style="vertical-align: -1px"> \sin x</math>&nbsp; is &nbsp; <math>\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.</math>
 |-
+|So, the power series of &nbsp;<math style="vertical-align: -5px">\sin(x^2)</math> &nbsp; is
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 |}

@@ Line 6: / Line 6: @@
 !Foundations: &nbsp;
 |-
+|What is the power series of &nbsp;<math style="vertical-align: -1px">\sin x?</math>
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; The power series of &nbsp;<math style="vertical-align: -1px"> \sin x</math>&nbsp; is &nbsp; <math>\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}.</math>
 |}

@@ Line 1: / Line 1: @@
-<span class="exam">(a) Express the indefinite integral &nbsp;<math>\int \sin(x^2)~dx</math>&nbsp; as a power series.
+<span class="exam">(a) Express the indefinite integral &nbsp;<math style="vertical-align: -13px">\int \sin(x^2)~dx</math>&nbsp; as a power series.
-<span class="exam">(b) Express the definite integral &nbsp;<math>\int_0^1 \sin(x^2)~dx</math>&nbsp; as a number series.
+<span class="exam">(b) Express the definite integral &nbsp;<math style="vertical-align: -14px">\int_0^1 \sin(x^2)~dx</math>&nbsp; as a number series.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+::<span class="exam">a) Express the indefinite integral <math>\int \sin(x^2)~dx</math> as a power series.
-::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>
+::<span class="exam">b) Express the definite integral <math>\int_0^1 \sin(x^2)~dx</math> as a number series.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"