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

From Grad Wiki
Jump to navigation Jump to search
 
(6 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
<span class="exam">Find the derivatives of the following functions. Do not simplify.
+
<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) <math>f(x)=x^3(x^{\frac{4}{3}}-1)</math>
+
::<span class="exam"><math>\text{BSA}=\frac{\sqrt{hm}}{60}</math>
−
::<span class="exam">b) <math>g(x)=\frac{x^3+x^{-3}}{1+6x}</math> where <math>x>0</math>
 
  
 +
<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?
 +
<hr>
 +
[[009A Sample Midterm 2, Problem 4 Solution|'''<u>Solution</u>''']]
  
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+
[[009A Sample Midterm 2, Problem 4 Detailed Solution|'''<u>Detailed Solution</u>''']]
−
!Foundations: &nbsp;
 
−
|-
 
−
|'''1.''' Product Rule
 
−
|-
 
−
|'''2.''' Quotient Rule
 
−
|}
 
  
  
−
'''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;
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|}
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|}
 
−
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Final Answer: &nbsp;
 
−
|-
 
−
|&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>
 
−
|-
 
−
|'''(b)'''
 
−
|}
 
 
[[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

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009A_Sample_Midterm_2,_Problem_4&oldid=3947"