Difference between revisions of "009A Sample Midterm 3, Problem 4"

Latest revision as of 14:24, 9 November 2017

Find the derivatives of the following functions. Do not simplify.

(a) $f(x)={\frac {(3x-5)(-x^{-2}+4x)}{x^{\frac {4}{5}}}}$

(b) $g(x)={\sqrt {x}}+{\frac {1}{\sqrt {x}}}+{\sqrt {\pi }}$ for $x>0.$

Solution

Detailed Solution

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@@ Line 1: / Line 1: @@
-<span class="exam">Find the equation of the tangent line to &nbsp;<math style="vertical-align: -4px">y=3\sqrt{-2x+5}</math>&nbsp; at &nbsp;<math style="vertical-align: -4px">(-2,9).</math>
+<span class="exam"> Find the derivatives of the following functions. Do not simplify.
+<span class="exam">(a)&nbsp; <math style="vertical-align: -16px">f(x)=\frac{(3x-5)(-x^{-2}+4x)}{x^{\frac{4}{5}}}</math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<span class="exam">(b)&nbsp; <math>g(x)=\sqrt{x}+\frac{1}{\sqrt{x}}+\sqrt{\pi}</math>&nbsp; for &nbsp;<math style="vertical-align: 0px">x>0.</math>
-!Foundations: &nbsp;
-|-
-|The equation of the tangent line to &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; at the point &nbsp;<math style="vertical-align: -5px">(a,b)</math>&nbsp; is
-|-
-|
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math style="vertical-align: -5px">y=m(x-a)+b</math>&nbsp; where &nbsp;<math style="vertical-align: -5px">m=f'(a).</math>
-|}
+<hr>
+[[009A Sample Midterm 3, Problem 4 Solution|'''<u>Solution</u>''']]
-'''Solution:'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009A Sample Midterm 3, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
-!Step 1: &nbsp;
-|-
-|First, we need to calculate the slope of the tangent line.
-|-
-|Let &nbsp;<math style="vertical-align: -5px">f(x)=3\sqrt{-2x+5}.</math>
-|-
-|From Problem 3, we have
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>f'(x)=-\frac{3}{\sqrt{-2x+5}}.</math>
-|-
-|Therefore, the slope of the tangent line is
-|-
-|
-&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
-\displaystyle{m} & = & \displaystyle{f'(-2)}\\
-&&\\
-& = & \displaystyle{-\frac{3}{\sqrt{-2(-2)+5}}}\\
-&&\\
-& = & \displaystyle{-\frac{3}{\sqrt{9}}}\\
-&&\\
-& = & \displaystyle{-1.}
-\end{array}</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
-|-
-|Now, the tangent line has slope &nbsp;<math style="vertical-align: -1px">m=-1</math>
-|-
-|and passes through the point &nbsp;<math style="vertical-align: -5px">(-2,9).</math>
-|-
-|Hence, the equation of the tangent line is
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>y=-1(x+2)+9.</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Final Answer: &nbsp;
-|-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>y=-1(x+2)+9</math>
-|-
-|
-|}
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009A Sample Midterm 3, Problem 4"

Latest revision as of 14:24, 9 November 2017

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