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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|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>  
 
|-
 
|-
 
|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 18:40, 26 February 2017

The rate of reaction to a drug is given by:

where  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t}   is the number of hours since the drug was administered.

Find the total reaction to the drug from  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle t=1}   to  


Foundations:  
If we calculate  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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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