Kayla Murray: /* Problem 7 */

2017-12-03T07:17:37Z

Problem 7

Kayla Murray: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 7B for the final. An actual test may or may not be similar.''' '''Click on the''' '''
2017-12-03T07:16:00Z

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'''This is a sample, and is meant to represent the material usually covered in Math 7B for the final. An actual test may or may not be similar.'''

'''Click on the''' ''' boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[007B_Sample Final 3,_Problem_1| Problem 1 ]] ==
Consider the region bounded by the following two function:

::<math>y=-x^2+9\text{ and }y=0.</math>

(a) Using the Left Hand Rule with three rectangles having equal width, approximate the area.

(b) Using the Right Hand Rule with three rectangles having equal width, approximate the area.

== [[007B_Sample Final 3,_Problem_2| Problem 2 ]] ==
 We would like to evaluate
::<math>\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t\,dt\bigg).</math>

(a) Compute  <math style="vertical-align: -15px">f(x)=\int_{-1}^{x} \sin(t^2)2t\,dt</math>.

(b) Find  <math style="vertical-align: -5px">f'(x)</math>.

(c) State the Fundamental Theorem of Calculus.

(d) Use the Fundamental Theorem of Calculus to compute  <math style="vertical-align: -15px">\frac{d}{dx}\bigg(\int_{-1}^{x} \sin(t^2)2t\,dt\bigg)</math>  without first computing the integral.

== [[007B_Sample Final 3,_Problem_3| Problem 3 ]] ==
Consider the area bounded by the following two functions:
::<math style="vertical-align: -4px">y=\sin x\text{ and }y=\frac{2}{\pi}x.</math>

== [[007B_Sample Final 3,_Problem_4| Problem 4 ]] ==
 Compute the following integrals.

(a)  <math>\int e^x(x+\sin(e^x))~dx</math>

(b)  <math>\int \frac{2x^2+1}{2x^2+x}~dx</math>

(c)  <math>\int \sin^3x~dx</math>

== [[007B_Sample Final 3,_Problem_5| Problem 5 ]] ==
Find the volume of the solid obtained by rotating the region bounded by the following about  <math style="vertical-align: 0px">x</math>-axis.

::<math>y=\sqrt{\sin x},~0\le x\le \pi,~y=0</math>

== [[007B_Sample Final 3,_Problem_6| Problem 6 ]] ==
 Evaluate the improper integrals:

(a)  <math>\int_0^{\infty} xe^{-x}~dx</math>

(b)  <math>\int_1^4 \frac{dx}{\sqrt{4-x}}</math>

== [[007B_Sample Final 1,_Problem_7| Problem 7 ]] ==

In a single-compartment model, the concentration of the solution at time  <math style="vertical-align: 0px">t,</math>  denoted by  <math style="vertical-align: 0px">C(t),</math>  is described by the differential equation:

::<math>\frac{dC}{dt}=\frac{0.2}{400} (3-C).</math>

Solve the above differential equation when  <math style="vertical-align: 0px">C(0)=0,</math>  and find  <math style="vertical-align: 0px">\displaystyle{\lim_{t\rightarrow \infty} C(t).</math>

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 == [[007B_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
-<span class="exam">In a single-compartment model, the concentration of the solution at time &nbsp;<math style="vertical-align: 0px">t,</math>&nbsp; denoted by &nbsp;<math style="vertical-align: 0px">C(t),</math>&nbsp; is described by the differential equation:
+<span class="exam">In a single-compartment model, the concentration of the solution at time &nbsp;<math style="vertical-align: -5px">t,</math>&nbsp; denoted by &nbsp;<math style="vertical-align: -5px">C(t),</math>&nbsp; is described by the differential equation:
 ::<math>\frac{dC}{dt}=\frac{0.2}{400} (3-C).</math>
-Solve the above differential equation when &nbsp;<math style="vertical-align: 0px">C(0)=0,</math>&nbsp; and find &nbsp;<math style="vertical-align: 0px">\displaystyle{\lim_{t\rightarrow \infty} C(t).</math>
+<span class="exam">Solve the above differential equation when &nbsp;<math style="vertical-align: -5px">C(0)=0,</math>&nbsp; and find &nbsp;<math style="vertical-align: -13px">\displaystyle{\lim_{t\rightarrow \infty} C(t)}.</math>