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

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::<math>r'(t)=2t^2e^{-t}</math>
 
::<math>r'(t)=2t^2e^{-t}</math>
  
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<span class="exam">where <math>t</math> is the number of hours since the drug was administered.  
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<span class="exam">where &nbsp;<math>t</math>&nbsp; is the number of hours since the drug was administered.  
  
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<span class="exam">Find the total reaction to the drug from <math style="vertical-align: -1px">t=1</math> to <math style="vertical-align: 0px">t=6.</math>
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<span class="exam">Find the total reaction to the drug from &nbsp;<math style="vertical-align: -1px">t=1</math>&nbsp; to &nbsp;<math style="vertical-align: 0px">t=6.</math>
  
  
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!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
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|-
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|If we calculate <math style="vertical-align: -14px">\int_a^b r'(t)~dt,</math> what are we calculating?
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|If we calculate &nbsp;<math style="vertical-align: -14px">\int_a^b r'(t)~dt,</math>&nbsp; what are we calculating?
 
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|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp; We are calculating <math style="vertical-align: -5px">r(b)-r(a).</math> This is the total reaction to the   
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&nbsp; &nbsp; &nbsp; &nbsp; We are calculating &nbsp;<math style="vertical-align: -5px">r(b)-r(a).</math>&nbsp; This is the total reaction to the   
 
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|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp; drug from <math style="vertical-align: 0px">t=a</math> to <math style="vertical-align: 0px">t=b.</math>  
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&nbsp; &nbsp; &nbsp; &nbsp; drug from &nbsp;<math style="vertical-align: 0px">t=a</math>&nbsp; to &nbsp;<math style="vertical-align: 0px">t=b.</math>  
 
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|}
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|-
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|To calculate the total reaction to the drug from <math style="vertical-align: -1px">t=1</math> to <math style="vertical-align: -4px">t=6,</math>  
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|To calculate the total reaction to the drug from &nbsp;<math style="vertical-align: -1px">t=1</math>&nbsp; to &nbsp;<math style="vertical-align: -4px">t=6,</math>  
 
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|-
 
|we need to calculate
 
|we need to calculate
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|We proceed using integration by parts.  
 
|We proceed using integration by parts.  
 
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|-
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|Let <math style="vertical-align: 0px">u=2t^2</math> and <math style="vertical-align: 0px">dv=e^{-t}dt.</math>  
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|Let &nbsp;<math style="vertical-align: 0px">u=2t^2</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">dv=e^{-t}dt.</math>  
 
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|-
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|Then, <math style="vertical-align: -1px">du=4t~dt</math> and <math style="vertical-align: 0px">v=-e^{-t}.</math>
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|Then, &nbsp;<math style="vertical-align: -1px">du=4t~dt</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">v=-e^{-t}.</math>
 
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|-
 
|Then, we have
 
|Then, we have
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|Now, we need to use integration by parts again.  
 
|Now, we need to use integration by parts again.  
 
|-
 
|-
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|Let <math style="vertical-align: 0px">u=4t</math> and <math style="vertical-align: 0px">dv=e^{-t}dt.</math>  
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|Let &nbsp;<math style="vertical-align: 0px">u=4t</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">dv=e^{-t}dt.</math>  
 
|-
 
|-
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|Then, <math style="vertical-align: -1px">du=4dt</math> and <math style="vertical-align: 0px">v=-e^{-t}.</math>
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|Then, &nbsp;<math style="vertical-align: -1px">du=4dt</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">v=-e^{-t}.</math>
 
|-
 
|-
 
|Thus, we get
 
|Thus, we get

Revision as of 17:40, 26 February 2017

The rate of reaction to a drug is given by:

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 t}   is the number of hours since the drug was administered.

Find the total reaction to the drug from  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 t=1}   to  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 t=6.}


Foundations:  
If we calculate  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 \int_a^b r'(t)~dt,}   what are we calculating?

        We are calculating  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 r(b)-r(a).}   This is the total reaction to the

        drug from  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 t=a}   to  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 t=b.}


Solution:

Step 1:  
To calculate the total reaction to the drug from  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 t=1}   to  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 t=6,}
we need to calculate

        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 \int_1^6 r'(t)~dt=\int_1^6 2t^2e^{-t}~dt.}

Step 2:  
We proceed using integration by parts.
Let  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 u=2t^2}   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 dv=e^{-t}dt.}
Then,  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 du=4t~dt}   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 v=-e^{-t}.}
Then, we have
       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 \int_1^62t^2e^{-t}~dt=\left. -2t^2e^{-t}\right|_1^6+\int_1^6 4te^{-t}~dt.}
Step 3:  
Now, we need to use integration by parts again.
Let  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 u=4t}   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 dv=e^{-t}dt.}
Then,  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 du=4dt}   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 v=-e^{-t}.}
Thus, we get

        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 \begin{array}{rcl} \displaystyle{\int_1^62t^2e^{-t}~dt} & = & \displaystyle{\left. -2t^2e^{-t}-4te^{-t}\right|_1^6+\int_1^6 4e^{-t}}\\ &&\\ & = & \displaystyle{\left. -2t^2e^{-t}-4te^{-t}-4e^{-t}\right|_1^6}\\ &&\\ & = & \displaystyle{-2(6)^2e^{-6}-4(6)e^{-6}-4e^{-6}}-(-2(1)^2e^{-1}-4(1)e^{-1}-4e^{-1}) \\ &&\\ & = & \displaystyle{\frac{-100+10e^5}{e^6}.} \end{array}}


Final Answer:  
       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 \frac{-100+10e^5}{e^6}}

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