Difference between revisions of "009B Sample Midterm 2"

Latest revision as of 11:13, 9 April 2017

This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.

Click on the boxed problem numbers to go to a solution.

Problem 1

This problem has three parts:

(a) State the Fundamental Theorem of Calculus.

(b) Compute ${\frac {d}{dx}}\int _{0}^{\cos(x)}\sin(t)~dt$ .

(c) Evaluate $\int _{0}^{\pi /4}\sec ^{2}x~dx$ .

Problem 2

Evaluate

(a) $\int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt$

(b) $\int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx$

Problem 3

A particle moves along a straight line with velocity given by:

v(t)=-32t+200

feet per second. Determine the total distance traveled by the particle

from time $t=0$ to time $t=10.$

Problem 4

Evaluate the integral:

\int e^{-2x}\sin(2x)~dx

Problem 5

Evaluate the integral:

\int \tan ^{4}x~dx

Contributions to this page were made by Kayla Murray

@@ Line 1: / Line 1: @@
-'''This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar. Click on the'''
+'''This is a sample, and is meant to represent the material usually covered in Math 9B for the midterm. An actual test may or may not be similar.'''
-'''<span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
+'''Click on the <span class="biglink" style="color:darkblue;">&nbsp;boxed problem numbers&nbsp;</span> to go to a solution.'''
 <div class="noautonum">__TOC__</div>
 == [[009B_Sample Midterm 2,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]] ==
-<span class="exam"> Consider the region <math style="vertical-align: 0px">S</math> bounded by <math style="vertical-align: -13px">x=1,x=5,y=\frac{1}{x^2}</math>&thinsp; and the <math>x</math>-axis.
+<span class="exam"> This problem has three parts:
+<span class="exam">(a) State the Fundamental Theorem of Calculus.
+<span class="exam">(b) Compute &nbsp; <math style="vertical-align: -15px">\frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt</math>.
-::<span class="exam">a) Use four rectangles and a Riemann sum to approximate the area of the region <math style="vertical-align: 0px">S</math>. Sketch the region <math style="vertical-align: 0px">S</math> and the rectangles and indicate whether your rectangles overestimate or underestimate the area of <math style="vertical-align: 0px">S</math>.
+<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -14px">\int_{0}^{\pi/4}\sec^2 x~dx</math>.
-::<span class="exam">b) Find an expression for the area of the region <math style="vertical-align: 0px">S</math> as a limit. Do not evaluate the limit.
 == [[009B_Sample Midterm 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
-<span class="exam"> This problem has three parts:
+<span class="exam"> Evaluate
-::<span class="exam">a) State the Fundamental Theorem of Calculus.
+<span class="exam">(a) &nbsp; <math style="vertical-align: -14px">\int_1^2\bigg(2t+\frac{3}{t^2}\bigg)\bigg(4t^2-\frac{5}{t}\bigg)~dt</math>
-::<span class="exam">b) Compute &thinsp; <math style="vertical-align: -15px">\frac{d}{dx}\int_0^{\cos (x)}\sin (t)~dt</math>.
+<span class="exam">(b) &nbsp; <math style="vertical-align: -14px">\int_0^2 (x^3+x)\sqrt{x^4+2x^2+4}~dx</math>
-::<span class="exam">c) Evaluate <math style="vertical-align: -14px">\int_{0}^{\pi/4}\sec^2 x~dx</math>.
+== [[009B_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+<span class="exam"> A particle moves along a straight line with velocity given by:
-== [[009B_Sample Midterm 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+::<math>v(t)=-32t+200</math>
-<span class="exam"> Evaluate:
-::<span class="exam">a) <math style="vertical-align: -14px">\int_1^2\bigg(2t+\frac{3}{t^2}\bigg)\bigg(4t^2-\frac{5}{t}\bigg)~dt</math>
+<span class="exam">feet per second. Determine the total distance traveled by the particle
-::<span class="exam">b) <math style="vertical-align: -14px">\int_0^2 (x^3+x)\sqrt{x^4+2x^2+4}~dx</math>
+<span class="exam">from time &nbsp;<math style="vertical-align: 0px">t=0</math>&nbsp; to time &nbsp;<math style="vertical-align: -1px">t=10.</math>
 == [[009B_Sample Midterm 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
@@ Line 35: / Line 38: @@
 ::<math>\int \tan^4 x ~dx</math>
 '''Contributions to this page were made by [[Contributors|Kayla Murray]]'''