Difference between revisions of "009A Sample Final 3, Problem 3"

Latest revision as of 16:36, 2 December 2017

Find the derivative of the following function using the limit definition of the derivative:

f(x)=3x-x^{2}

Solution

Detailed Solution

Return to Sample Exam

@@ Line 3: / Line 3: @@
 ::<span class="exam"><math>f(x)=3x-x^2</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009A Sample Final 3, Problem 3 Solution|'''<u>Solution</u>''']]
-|-
-|<math style="vertical-align: -13px">f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}</math>
-|}
-'''Solution:'''
+[[009A Sample Final 3, Problem 3 Detailed Solution|'''<u>Detailed Solution</u>''']]
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|Using the limit definition of derivative, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{f'(x)} & = & \displaystyle{\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}}\\
-&&\\
-& = & \displaystyle{\lim_{h\rightarrow 0} \frac{(3(x+h)-(x+h)^2)-(3x-x^2)}{h}}\\
-&&\\
-& = & \displaystyle{\lim_{h\rightarrow 0} \frac{3x+3h-(x^2+2xh+h^2)-3x+x^2}{h}}\\
-&&\\
-& = & \displaystyle{\lim_{h\rightarrow 0} \frac{3h-2xh-h^2}{h}.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{f'(x)} & = & \displaystyle{\lim_{h\rightarrow 0} \frac{3h-2xh-h^2}{h}}\\
-&&\\
-& = & \displaystyle{\lim_{h\rightarrow 0} 3-2x-h}\\
-&&\\
-& = & \displaystyle{3-2x.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>f'(x)=3-2x</math>
-|}
 [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Final 3, Problem 3"

Latest revision as of 16:36, 2 December 2017

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