Difference between revisions of "008A Sample Final A, Question 12"

From Math Wiki
Jump to navigation Jump to search
(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"...")
 
 
(One intermediate revision by the same user not shown)
Line 2: Line 2:
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations
+
!Foundations: &nbsp;
 
|-
 
|-
 
|1) f(x + h) = ?
 
|1) f(x + h) = ?
Line 10: Line 10:
 
|Answer:
 
|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 18: Line 18:
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| 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>
 
|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;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
+
!Step 2: &nbsp;
 
|-
 
|-
 
|Now we simplify the numerator:  
 
|Now we simplify the numerator:  
 
|- style = "text-align:center;"
 
|- 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>
 
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Arithmetic:
+
!Step 3: &nbsp;
 
|-
 
|-
 
|Now we simplify the numerator:  
 
|Now we simplify the numerator:  
 
|- style = "text-align:center;"
 
|- 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))}
+
& & \\
 +
& = & \displaystyle{\frac{-6}{(3(x + h) + 1)(3x + 1))}}
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
+
!Final Answer: &nbsp;
 
|-
 
|-
 
|<math>\frac{-6}{(3(x + h) + 1)(3x + 1))}</math>
 
|<math>\frac{-6}{(3(x + h) + 1)(3x + 1))}</math>
 
|}
 
|}
 
[[008A Sample Final A|<u>'''Return to Sample Exam</u>''']]
 
[[008A Sample Final A|<u>'''Return to Sample Exam</u>''']]

Latest revision as of 23:59, 25 May 2015

Question: Find and simplify the difference quotient for f(x) =

Foundations:  
1) f(x + h) = ?
2) How do you eliminate the 'h' in the denominator?
Answer:
1).
2) The numerator of the difference quotient is   so the first step is to simplify this expression. This then allows us to eliminate the 'h' in the denominator.

Solution:

Step 1:  
The difference quotient that we want to simplify is
Step 2:  
Now we simplify the numerator:
Step 3:  
Now we simplify the numerator:
Final Answer:  

Return to Sample Exam