008A Sample Final A, Question 12 - Revision history

MathAdmin at 06:59, 26 May 2015

2015-05-26T06:59:37Z

MathAdmin at 23:04, 23 May 2015

2015-05-23T23:04:45Z

MathAdmin: Created page with "'''Question: ''' Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for f(x) = $\frac{2}{3x+1}$ {| class="mw-collapsible mw-collapsed"..."

2015-04-28T21:12:17Z

Created page with "'''Question: ''' Find and simplify the difference quotient <math>\frac{f(x+h)-f(x)}{h}</math> for f(x) = <math>\frac{2}{3x+1}</math> {| class="mw-collapsible mw-collapsed"..."

New page

'''Question: ''' Find and simplify the difference quotient <math>\frac{f(x+h)-f(x)}{h}</math> for f(x) = <math>\frac{2}{3x+1}</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations
|-
|1) f(x + h) = ?
|-
|2) How do you eliminate the 'h' in the denominator?
|-
|Answer:
|-
|1) Since <math>f(x + h) = \frac{2}{3(x + h) + 1}</math> the difference quotient is a difference of fractions divided by h.
|-
|2) The numerator is <math>\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}</math> so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.
|}

Solution:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
|-
|The difference quotient that we want to simplify is <math>\frac{f(x + h) - f(x)}{h} = \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
|-
|Now we simplify the numerator:
|- style = "text-align:center;"
|
<math>\begin{array}{rcl}
\frac{f(x + h) - f(x)}{h} &=& \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h\\

&=& \frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Arithmetic:
|-
|Now we simplify the numerator:
|- style = "text-align:center;"
|
<math>\begin{array}{rcl}
\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))} & = & \frac{6x + 2 - 6x -6h -2}{h(3(x + h) + 1)(3x + 1))}\\
& = & \frac{-6}{(3(x + h) + 1)(3x + 1))}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
|-
|<math>\frac{-6}{(3(x + h) + 1)(3x + 1))}</math>
|}
[[008A Sample Final A|<u>'''Return to Sample Exam</u>''']]

@@ Line 2: / Line 2: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Foundations
+!Foundations: &nbsp;
 |-
 |1) f(x + h) = ?
@@ Line 18: / Line 18: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Step 1:
+!Step 1: &nbsp;
 |-
 |The difference quotient that we want to simplify is <math>\frac{f(x + h) - f(x)}{h} = \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h</math>
@@ Line 24: / Line 24: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Step 2:
+!Step 2: &nbsp;
 |-
 |Now we simplify the numerator:
@@ Line 37: / Line 37: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Arithmetic:
+!Step 3: &nbsp;
 |-
 |Now we simplify the numerator:
@@ Line 50: / Line 50: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answer:
+!Final Answer: &nbsp;
 |-
 |<math>\frac{-6}{(3(x + h) + 1)(3x + 1))}</math>
 |}
 [[008A Sample Final A|<u>'''Return to Sample Exam</u>''']]

@@ Line 10: / Line 10: @@
 |Answer:
 |-
-|1) Since <math>f(x + h) = \frac{2}{3(x + h) + 1}</math> the difference quotient is a difference of fractions divided by h.
+|1)<math>f(x + h) = \frac{2}{3(x + h) + 1}</math>.
 |-
-|2) The numerator is <math>\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}</math> so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.
+|2) The numerator of the difference quotient is <math>\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}</math>&nbsp; so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.
 |}
@@ Line 29: / Line 29: @@
 |- style = "text-align:center;"
+|
-<math>\begin{array}{rcl}
+::<math>\begin{array}{rcl}
-\frac{f(x + h) - f(x)}{h} &=& \left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h\\
+\displaystyle{\frac{f(x + h) - f(x)}{h}} &=& \displaystyle{\left(\frac{2}{3(x + h) + 1} - \frac{2}{3x + 1}\right) \div h}\\
+& & \\
-&=& \frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}
+&=& \displaystyle{\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}}
 \end{array}</math>
 |}
@@ Line 42: / Line 42: @@
 |- style = "text-align:center;"
+|
-<math>\begin{array}{rcl}
+::<math>\begin{array}{rcl}
-\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))} & = & \frac{6x + 2 - 6x -6h -2}{h(3(x + h) + 1)(3x + 1))}\\
+\displaystyle{\frac{2(3x + 1) -2(3(x + h) + 1)}{h(3(x + h) + 1)(3x + 1))}} & = & \displaystyle{\frac{6x + 2 - 6x -6h -2}{h(3(x + h) + 1)(3x + 1))}}\\
-& = & \frac{-6}{(3(x + h) + 1)(3x + 1))}
+& & \\
 \end{array}</math>
 |}

008A Sample Final A, Question 12 - Revision history

MathAdmin at 06:59, 26 May 2015

MathAdmin at 23:04, 23 May 2015

MathAdmin: Created page with "'''Question: ''' Find and simplify the difference quotient \frac{f(x+h)-f(x)}{h} for f(x) = \frac{2}{3x+1} {| class="mw-collapsible mw-collapsed"..."

MathAdmin: Created page with "'''Question: ''' Find and simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ for f(x) = $\frac{2}{3x+1}$ {| class="mw-collapsible mw-collapsed"..."