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:
  
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<span class="exam">(a) <math>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>
  
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<span class="exam">(b) <math>g(x)=\frac{x^3+x^{-3}}{1+6x}</math> where <math>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>''']]
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Foundations: &nbsp;
 
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|-
 
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|'''1.''' '''Product Rule'''
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)</math>
 
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|-
 
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|'''2.''' '''Quotient Rule'''
 
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|-
 
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|&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>
 
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|-
 
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|'''3.''' '''Power Rule'''
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{d}{dx}(x^n)=nx^{n-1}</math>
 
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|}
 
  
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'''Solution:'''
 
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'''(a)'''
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|-
 
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|Using the Product Rule, we have
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>f'(x)=x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1).</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|-
 
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|Now, we have
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{f'(x)} & = & \displaystyle{x^3(x^{\frac{4}{3}}-1)'+(x^3)'(x^{\frac{4}{3}}-1)}\\
 
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&&\\
 
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& = & \displaystyle{x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1).}
 
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\end{array}</math>
 
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|}
 
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'''(b)'''
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;
 
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|-
 
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|Using the Quotient Rule, we have
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>g'(x)=\frac{(1+6x)(x^3+x^{-3})'-(x^3+x^{-3})(1+6x)'}{(1+6x)^2}.</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
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|-
 
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|Now, we have
 
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|-
 
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|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{g'(x)} & = & \displaystyle{\frac{(1+6x)(x^3+x^{-3})'-(x^3+x^{-3})(1+6x)'}{(1+6x)^2}}\\
 
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&&\\
 
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& = & \displaystyle{\frac{(1+6x)(3x^2-3x^{-4})-(x^3+x^{-3})(6)}{(1+6x)^2}.}
 
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\end{array}</math>
 
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Final Answer: &nbsp;
 
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|-
 
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|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>x^3\bigg(\frac{4}{3}x^{\frac{1}{3}}\bigg)+(3x^2)(x^{\frac{4}{3}}-1)</math>
 
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|-
 
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|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{(1+6x)(3x^2-3x^{-4})-(x^3+x^{-3})(6)}{(1+6x)^2}</math>
 
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|}
 
 
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]
 
[[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 14:39, 9 November 2017

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{BSA}=\frac{\sqrt{hm}}{60}}

where  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h}   is the height in centimeters and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m}   is the mass in kilograms. Calculate the rate of change of BSA with respect to height for a person of a constant mass of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=85.}   What is the rate at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=170}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=190?}   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

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