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Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Exam 2 Review
Exam 2 Review
Directions.
The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
An experiment consists of rolling a fair, four-sided die and observing the number that lands uppermost. If the result is an even number, we draw a card from a standard well-shuffled 52 card deck and record the color. If the result is odd, we flip a fair coin and record whether the side that lands uppermost is heads or tails.
Let \(H = \) heads, \(T= \)tails, \(R= \) Red, and \(B= \) Black
Find an appropriate sample space for this experiment.
Write (or list) the events:
\(E\) is the event a red card is drawn.
\(F\) is the event a tails occurs.
\(G\) is the event an even number is rolled.
List and describe the event \(E \cup F^C \cap G\).
Answer
\(S=\{(1,H), (1, T), (2, R), (2, B), (3, H), (3, T), (4, R), (4,B)\}\)
\(E=\{(2, R), (4, R)\}\)
\(F=\{(1, T), (3, T)\}\)
\(G=\{(2, R), (2, B), (4, R), (4, B)\}\)
\(E \cup F^C \cap G= G\)
Video
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MATH 140 WIR6 #1
Before Thanksgiving a group of Aggies were surveyed and asked to choose their favorite side
dish at Thanksgiving dinner. The survey results are displayed in the table below.
Candied
Yams
Corn
Bread
Dressing
Roasted
Brussels
Sprouts
Green
Beans
Other
Total
Freshman
5
20
30
10
20
85
Sophomore
7
14
17
11
6
55
Junior
2
32
25
20
10
89
Senior
7
19
30
7
8
71
Total
21
85
102
48
44
300
If one Aggie is randomly selected from this group, what is the probability that the Aggie:
likes roasted Brussels sprouts?
is a freshman?
is a sophomore whose favorite side dish was green beans?
is a senior or junior whose favorite side dish was candied yams?
Answer
\(\dfrac{102}{300}\)
\(\dfrac{85}{300}\)
\(\dfrac{11}{300}\)
\(\dfrac{89+71+5+7}{300} = \dfrac{172}{300}\)
Video
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MATH 140 WIR6 #2
Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose that \(P(E) = 0.4, P(F) = 0.3,\) and \(P(E \cup F) = 0.6\). Find \(P(E^C \cap F)\).
Answer
\(P(E^C \cap F) = 0.2\)
Video
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MATH 140 WIR6 #3
An experiment has a sample space of \(S = \{i,\ l,\ o,\ v,\ e,\ m,\ a,\ t,\ h\}\), with events \(E = \{t, a, i, l\} \), \(F = \{l,\ o,\ v,\ e\}\), and \(G=\{m,\ a,\ t,\ h\}\). Find each of the following.
\(E \cup F^C\)
\(E \cap F^C\)
\((F \cap G)^C\)
Answer
\(E \cup F^C = \{t, a, i, l, m, h\}\)
\(E \cap F^C = \{t, a, i\}\)
\((F \cap G)^C = S\)
Video
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MATH 140 WIR6 #4
Let \(S = \{s_1, s_2, s_3, s_4, s_5\}\) be the sample space associated with the following
partial
probability distribution table.
\[
\begin{array}{l|c|c|c|c|c}
\text{Outcome} & s_1 & s_2 & s_3 & s_4 & s_5\\
\hline
\text{Probability} & \dfrac{3}{20} & \dfrac{7}{20} & \quad & \dfrac{4}{20} & \dfrac{1}{20}
\end{array}
\]Let \(E = \{s_3, s_4\}, F=\{ s_4, s_5\}\), and \(G=\{s_1, s_2, s_3\}\), find:
\(P(E)\)
\(P(F^C)\)
\(P(E \cap G)\)
\(P(G \cup E)\)
\(P(F \cap G)\)
Answer
\(\dfrac{9}{20}\)
\(\dfrac{15}{20} \)
\(\dfrac{5}{20}\)
\(\dfrac{19}{20}\)
\(0\)
Video
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MATH 140 WIR6 #5
An experiment consists of rolling two distinguishable four-sided die and observing the two numbers that land uppermost.
Let \(Y\) be the random variable denoting the number of fours that land uppermost on the dice. Create the probability distribution for \(Y\).
Let \(X\) be the random variable denoting the product of the two numbers. Create the
probability distribution for \(X\) and then find the expected value for \(X\).
Answer
\(\begin{array}{c|c c c}
Y & 0 & 1 & 2 \\[3pt]
\hline
P(Y) & \dfrac{9}{16} & \dfrac{6}{16} & \dfrac{1}{16}\\[5pt]
\end{array}\)
\(\begin{array}{c|c c c c c c c c c}
X & 1 & 2 & 3 & 4 & 6 & 8 & 9 & 12 & 16 \\[5pt]
\hline
P(X) & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{3}{16} & \dfrac{2}{16} & \dfrac{2}{16} & \dfrac{1}{16} & \dfrac{2}{16} & \dfrac{1}{16}\\[5pt]
\end{array}\)
\(E(X) = 6.25\)
Video
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MATH 140 WIR6 #6
Use the given
partial
histogram below, to answer the questions that follow.
Compute the value of \(P(X = 4.)\)
Compute the value of \(P(X \leq 2 ).\)
Compute the value of \(P(-2 < X \leq 4).\)
Compute the value of \(P(X \geq 0).\)
Calculate \(E(X).\)
Answer
\(P(X = 4) = 0.2\)
\(P(X \leq 2 ) = 0.45\)
\(P(-2 < X \leq 4) = 0.3 + 0.1 + 0.2 = 0.6\)
\(P(X \geq 0) = 1 - 0.05 = 0.95\)
\(E(X) = -2(0.05) + 0(0.3)+ 2(0.1) + 4(0.2) + 6(0.35) = 3\)
Video
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MATH 140 WIR6 #7
For the linear programming problem below, define the variables, set up the objective function and constraints, then determine if it is a standard maximization problem.
A company manufactures Products A, B, and C. Each product is processed in three departments: Dept.
I, Dept. II, and Dept. III. The total available labor-hours, per week, for departments I, II, and III are
800 hours, 1080 hours, and 840 hours, respectively. The number of labor-hours required to manufacture each product in each department, as well as the profit realized per product for each of the three products are displayed in the table below.
Product A
Product B
Product C
Dept. I
3 hours
1 hour
3 hours
Dept. II
4 hours
2 hours
2 hours
Dept. III
2 hours
3 hours
1 hour
Profit
$16
$20
$15
How many units of each product should the company produce each week to maximize their profit?
Answer
Let x by the number of Product A manufactured each week.
Let y by the number of Product B manufactured each week.
Let z by the number of Product C manufactured each week.
Objective function: \(P = 16x + 20y + 15z\)
Constraints: \[
\begin{cases}
3x + y + z \leq 800 \\
4x + 2y + 2z \leq 1080 \\
2x + 3y + z \leq 840 \\
x, y, z \geq 0
\end{cases}
\]
Yes, standard max.
Video
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MATH 140 WIR6 #9