Difference between revisions of "009A Sample Midterm 2, Problem 4"

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<span class="exam">Find the derivatives of the following functions. Do not simplify.
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<span class="exam"> To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:
  
<span class="exam">(a) &nbsp; <math style="vertical-align: -5px">f(x)=x^3(x^{\frac{4}{3}}-1)</math>
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::<span class="exam"><math>\text{BSA}=\frac{\sqrt{hm}}{60}</math>
  
<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">g(x)=\frac{x^3+x^{-3}}{1+6x}</math>&nbsp; where &nbsp;<math style="vertical-align: 0px">x>0</math>
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<span class="exam">where &nbsp;<math style="vertical-align: 0px">h</math>&nbsp; is the height in centimeters and &nbsp;<math style="vertical-align: 0px">m</math>&nbsp; is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of &nbsp;<math style="vertical-align: 0px">m=85.</math>&nbsp; What is the rate at &nbsp;<math style="vertical-align: -1px">h=170</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">h=190?</math>&nbsp; Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?
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<hr>
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[[009A Sample Midterm 2, Problem 4 Solution|'''<u>Solution</u>''']]
  
  
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[[009A Sample Midterm 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
  
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;
 
|-
 
|'''1.''' '''Product Rule'''
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)</math>
 
|-
 
|'''2.''' '''Quotient Rule'''
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}\bigg(\frac{f(x)}{g(x)}\bigg)=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}</math>
 
|-
 
|'''3.''' '''Power Rule'''
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(x^n)=nx^{n-1}</math>
 
|}
 
  
 
'''Solution:'''
 
 
'''(a)'''
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;
 
|-
 
|Using the Product Rule, we have
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>f'(x)=x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1).</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{x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1)}\\
 
&&\\
 
& = & \displaystyle{x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1).}
 
\end{array}</math>
 
|}
 
 
'''(b)'''
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;
 
|-
 
|Using the Quotient Rule, we have
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>g'(x)=\frac{(1+6x)(x^3+x^{-3})'-(x^3+x^{-3})(1+6x)'}{(1+6x)^2}.</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{g'(x)} & = & \displaystyle{\frac{(1+6x)(x^3+x^{-3})'-(x^3+x^{-3})(1+6x)'}{(1+6x)^2}}\\
 
&&\\
 
& = & \displaystyle{\frac{(1+6x)(3x^2-3x^{-4})-(x^3+x^{-3})(6)}{(1+6x)^2}.}
 
\end{array}</math>
 
|}
 
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Final Answer: &nbsp;
 
|-
 
|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>f'(x)=x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1)</math>
 
|-
 
|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>g'(x)=\frac{(1+6x)(3x^2-3x^{-4})-(x^3+x^{-3})(6)}{(1+6x)^2}</math>
 
|}
 
 
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 12:08, 11 November 2017

To determine drug dosages, doctors estimate a person's body surface area (BSA) (in meters squared) using the formula:

where    is the height in centimeters and    is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of    What is the rate at    and    Express your results in the correct units. Does the BSA increase more rapidly with respect to height at lower or higher heights?


Solution


Detailed Solution


Return to Sample Exam