Difference between revisions of "009B Sample Midterm 1, Problem 3"

Latest revision as of 10:27, 20 November 2017

A population grows at a rate

P'(t)=500e^{-t}

where $P(t)$ is the population after $t$ months.

(a) Find a formula for the population size after $t$ months, given that the population is $2000$ at $t=0.$

(b) Use your answer to part (a) to find the size of the population after one month.

Solution

Detailed Solution

Return to Sample Exam

@@ Line 1: / Line 1: @@
-<span class="exam">Evaluate the indefinite and definite integrals.
+<span class="exam"> A population grows at a rate
-::<span class="exam">a) <math>\int x^2 e^xdx</math>
+::<math>P'(t)=500e^{-t}</math>
-::<span class="exam">b) <math>\int_{1}^{e} x^3\ln x~dx</math>
+<span class="exam">where &nbsp;<math style="vertical-align: -5px">P(t)</math>&nbsp; is the population after &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; months.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<span class="exam">(a) &nbsp; Find a formula for the population size after &nbsp;<math style="vertical-align: 0px">t</math>&nbsp; months, given that the population is &nbsp;<math style="vertical-align: 0px">2000</math>&nbsp; at &nbsp;<math style="vertical-align: 0px">t=0.</math>
-!Foundations: &nbsp;
-|-
-|Review integration by parts
-|}
-'''Solution:'''
+<span class="exam">(b) &nbsp; Use your answer to part (a) to find the size of the population after one month.
+<hr>
+[[009B Sample Midterm 1, Problem 3 Solution|'''<u>Solution</u>''']]
-'''(a)'''
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
-|-
-|We proceed using integration by parts. Let <math>u=x^2</math> and <math>dv=e^xdx</math>. Then, <math>du=2xdx</math> and <math>v=e^x</math>.
-|-
-|Therefore, we have
-|-
-|<math>\int x^2 e^xdx=x^2e^x-\int 2xdx</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009B Sample Midterm 1, Problem 3 Detailed Solution|'''<u>Detailed Solution</u>''']]
-!Step 2: &nbsp;
-|-
-|Now, we need to use integration by parts again. Let <math>u=2x</math> and <math>dv=e^xdx</math>. Then, <math>du=2dx</math> and <math>v=e^x</math>.
-|-
-|Therefore, we have
-|-
-|<math>\int x^2 e^xdx=x^2e^x-\bigg(2xe^x-\int 2e^xdx\bigg)=x^2e^x-2xe^x+2e^x+C</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;
-|-
-|'''(a)''' <math>x^2e^x-2xe^x+2e^x+C</math>
-|-
-|'''(b)'''
-|}
 [[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Difference between revisions of "009B Sample Midterm 1, Problem 3"

Latest revision as of 10:27, 20 November 2017

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