009A Sample Final 3, Problem 8 - Revision history

Kayla Murray at 23:56, 2 December 2017

2017-12-02T23:56:56Z

Kayla Murray at 20:45, 18 March 2017

2017-03-18T20:45:19Z

Kayla Murray at 18:57, 7 March 2017

2017-03-07T18:57:02Z

Kayla Murray at 18:40, 7 March 2017

2017-03-07T18:40:01Z

Kayla Murray at 21:39, 6 March 2017

2017-03-06T21:39:44Z

Kayla Murray at 21:36, 6 March 2017

2017-03-06T21:36:18Z

Kayla Murray at 02:03, 19 February 2017

2017-02-19T02:03:21Z

Kayla Murray at 02:35, 18 February 2017

2017-02-18T02:35:01Z

Kayla Murray: Created page with "Compute ::a) $\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}$ ::b)
2017-02-18T02:16:29Z

Created page with "Compute ::a) <math style="vertical-align: -14px">\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}</math> ::b) <math st..."

New page
Compute

::a) <math style="vertical-align: -14px">\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}</math>

::b) <math style="vertical-align: -14px">\lim_{x\rightarrow 0} \frac{\sin^2x}{3x}</math>

::c) <math style="vertical-align: -14px">\lim_{x\rightarrow -\infty} \frac{\sqrt{x^2+2}}{2x-1}</math>

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'''(a)'''

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'''(b)'''

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'''(c)'''

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[[009A_Sample_Final_3|'''Return to Sample Exam''']]

← Older revision		Revision as of 23:56, 2 December 2017
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−	<span class="exam">~~Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure~~  <math style="vertical-align: 0px">P</math>  and ~~volume~~  <math style="vertical-align: 0px">V</math>  ~~satisfy~~ the ~~equation  <math style="vertical-align: 0px">PV=C</math>  where  <math style="vertical-align: 0px">C</math>  is a constant. Suppose that at a certain instant~~, ~~the volume is~~  <math style="vertical-align: -~~4px~~">~~600~~ \~~text~~{ cm}~~^3,</math>  the pressure is  <math style="vertical-align: -4px">150 \text~~{ ~~kPa~~},</math>  ~~and~~ the ~~pressure~~ is ~~increasing at~~ a rate of ~~ ~~<~~math style="vertical-align: -4px"~~>~~20 \text{ kPa/min}.~~</~~math~~>~~  At what rate is the volume decreasing at this instant?~~	+	<span class="exam">If  <math style="vertical-align: 0px">W</math>  denotes the weight in pounds of an individual, and  <math style="vertical-align: 0px">t</math>  denotes the time in months, then  <math style="vertical-align: -13px">\frac{dW}{dt}</math>  is the rate of weight gain or loss in lbs/mo. The current speed record for weight loss is a drop in weight from 487 pounds to 130 pounds over an eight month period. Show that the rate of weight loss exceeded 44 lbs/mo at some time during the eight month period.
		+	<hr>
		+	[[009A Sample Final 3, Problem 8 Solution\|'''<u>Solution</u>''']]

−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Foundations:  ~~
−	\|-
−	~~\|'''Product Rule'''~~
−	\|-
−	~~\|        <math>\frac{d}{dx}(f(x)g(x))=f(x)g'(x)+f'(x)g(x)</math>~~
−	\|}

		+	[[009A Sample Final 3, Problem 8 Detailed Solution\|'''<u>Detailed Solution</u>''']]

−	~~'''Solution:'''~~

−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 1:  ~~
−	\|-
−	~~\|First, we take the derivative of the equation  <math style="vertical-align: 0px">PV=C.</math>~~
−	\|-
−	~~\|Using the product rule, we get~~
−	\|-
−	~~\|       <math>P'V+PV'=C'.</math>~~
−	\|-
−	~~\|Since  <math style="vertical-align: 0px">C</math>  is a constant,~~
−	\|-
−	~~\|       <math style="vertical-align: -1px">C'=0.</math> ~~
−	\|-
−	~~\|Therefore, we have~~
−	\|-
−	~~\|       <math>P'V+PV'=0.</math>~~
−	\|}
−
−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Step 2:  ~~
−	\|-
−	~~\|Solving for  <math style="vertical-align: -4px">V',</math>  we get~~
−	\|-
−	~~\|       <math>V'=\frac{-P'V}{P}.</math>~~
−	\|-
−	~~\|Using the information provided in the problem, we have~~
−	\|-
−	~~\|       <math style="vertical-align: 0px">V=600 \text{ cm}^3,~P=150 \text{ kPa},~P'=20 \text{ kPa/min}.</math> ~~
−	\|-
−	~~\|Hence, we get~~
−	\|-
−	\|
−	~~        <math>\begin{array}{rcl}~~
−	~~\displaystyle{V'} & = & \displaystyle{\frac{-(20)(600)}{150} \text{ cm}^3\text{/min}}\\~~
−	~~&&\\~~
−	~~& = & \displaystyle{-80 \text{ cm}^3\text{/min}.}~~
−	~~\end{array}</math>~~
−	\|-
−	~~\|Therefore, the volume is decreasing at a rate of  <math style="vertical-align: -5px">80 \text{ cm}^3\text{/min}</math>  at this instant.~~
−	\|}
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−	~~{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"~~
−	~~!Final Answer:  ~~
−	\|-
−	~~\|       The volume is decreasing at a rate of  <math style="vertical-align: -5px">80 \text{ cm}^3\text{/min}</math>  at this instant.~~
−	\|}
	[[009A_Sample_Final_3\|'''<u>Return to Sample Exam</u>''']]		[[009A_Sample_Final_3\|'''<u>Return to Sample Exam</u>''']]

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 |&nbsp; &nbsp; &nbsp; &nbsp;<math>P'V+PV'=C'.</math>
 |-
-|Since &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is a constant, &nbsp;<math style="vertical-align: -1px">C'=0.</math>&nbsp;
+|Since &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is a constant,
 |-
 |Therefore, we have

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 !Foundations: &nbsp;
 |-
+|'''Product Rule'''
 |-
+|&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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 '''Solution:'''
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 |}
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-!Step 1: &nbsp;
+!Final Answer: &nbsp;
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+|
 |}
 [[009A_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 21:36, 6 March 2017
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−	<span class="exam">Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure <math>P</math> and volume <math>V</math> satisfy the equation <math>PV=C</math> where <math>C</math> is a constant. Suppose that at a certain instant, the volume is <math>600 \text{ cm}^3,</math> the pressure is <math>150 \text{ kPa},</math> and the pressure is increasing at a rate of <math>20 \text{ kPa/min}.</math> At what rate is the volume decreasing at this instant?	+	<span class="exam">Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure  <math style="vertical-align: 0px">P</math>  and volume  <math style="vertical-align: 0px">V</math>  satisfy the equation  <math style="vertical-align: 0px">PV=C</math>  where  <math style="vertical-align: 0px">C</math>  is a constant. Suppose that at a certain instant, the volume is  <math style="vertical-align: -4px">600 \text{ cm}^3,</math>  the pressure is  <math style="vertical-align: -4px">150 \text{ kPa},</math>  and the pressure is increasing at a rate of  <math style="vertical-align: -4px">20 \text{ kPa/min}.</math>  At what rate is the volume decreasing at this instant?

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

← Older revision		Revision as of 02:35, 18 February 2017
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−	<span class="exam">~~Compute~~	+	<span class="exam">Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure <math>P</math> and volume <math>V</math> satisfy the equation <math>PV=C</math> where <math>C</math> is a constant. Suppose that at a certain instant, the volume is <math>600 \text{ cm}^3,</math> the pressure is <math>150 \text{ kPa},</math> and the pressure is increasing at a rate of <math>20 \text{ kPa/min}.</math> At what rate is the volume decreasing at this instant?
−
−	::<~~span class="exam"~~>a) <math ~~style="vertical-align: -14px"~~>~~\lim_{x\rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}~~</math>
−
−	::<~~span class~~=~~"exam"~~>b) <math ~~style="vertical-align: -14px"~~>\~~lim_~~{~~x\rightarrow 0~~} ~~\frac{\sin~~^~~2x}{3x}~~</math>
−
−	~~::<span class="exam">c)~~ <math ~~style="vertical-align: -14px"~~>\~~lim_~~{~~x\rightarrow -\infty~~} \~~frac~~{~~\sqrt{x^2+2}}{2x-1~~}</math>

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