# Mathematicsfor Managerial, Life and Social SciencesUniversity PreparationePrep Course ### Main Course Contents

1. FUNDAMENTALS OF ALGEBRA.
2. FUNCTIONS AND THEIR GRAPHS.
3. EXPONENTIAL AND LOGARITHMIC FUNCTIONS.
4. MATHEMATICS OF FINANCE.
5. SYSTEMS OF LINEAR EQUATIONS AND MATRICES
6. LINEAR PROGRAMMING.
7. SETS AND PROBABILITY.
9. THE DERIVATIVE.
10. APPLICATIONS OF THE DERIVATIVE.
11. INTEGRATION.
12. CALCULUS OF SEVERAL VARIABLES.

The details of the topics are given here.

Learning materials for all chapters are provided, but for the purpose of certification, a student has to pass the tests for the first three chapters and any other three from the remaining nine chapters.

### 1. Video Lesson (Logarithmic Function)

This short video lesson illustrates the relationship between base and exponent and the resulting number expressed in exponential form or logarithmic form.

##### 2. Question and Answer (Linear Programming)

Question: A financier plans to invest up to \$500,000 in two projects. Project A yields a return of 10% on the investment whereas project B yields a return of 15% on the investment. Because investment in project B is riskier than the investment in project A, the financier has decided that the investment in project B should not exceed 40% of the total investment.  How much should she invest in each project in order to maximize the return on her investment?

Answer: Let x and y denote the amount (in thousands of dollars) to be invested in project A and project B, respectively. Since the amount available for investment is up to \$500,000, we have

x + y 500

Next, the condition on the allocation of the funds implies that

y 0.4(x + y),  –0.4x + 0.6y ≤ 0, or  –2x + 3y 0

The linear programming problem at hand is

Maximize P = 0.1x + 0.15y subject to

x + y 500

–2x + 3y 0

x ≥ 0, y ≥ 0

### Samples of Bonus Materials

##### 1. Video Lesson on Business Finance (Intrinsic Value Vs Market Value)

This short video explains the differences between market price and intrinsic value of an asset such as stock and bond.

More on Business Finance ePrep Course

##### 2. Video Lesson on Physics (Positions and Velocities)

This short video lesson illustrates how velocity-time graph can be derived from position-time graph.

More on Physics ePrep Course

##### 3. Question and Solution on Probability & Statistics (Probability Distribution)

Question: Suppose that the unemployment rate in a given community is 7%. Four households are randomly selected to be interviewed. In each household, it is determined whether or not the primary wage earner is unemployed. If the 7% rate is correct, find the probability distribution for x, the number of primary wage earners who are unemployed.

Answer: Let U be the event that the primary wage earner is unemployed and let E be the event that the primary wage earner is employed. There are 16 simple events with unequal probabilities.

p(0) = P[E∩E∩E∩E] = (.93)4 = .7481

p(1) = 4P(E)3P(U) = 4 (.93)3 (.07) = .2252

p(2) = 6P(E)2P(U)2 = 6 (.93)(.07)2 = .0254

p(3) = 4P(E)P(U)3 = 4 (.93) (.07)3 = .0013

p(4) = P[U∩U∩U∩U] = (.07)4 = .000024

More on Statistics ePrep Course

##### 4. Video Lesson on Corporate Finance (Pricing of an Asset)

This video lesson discusses the fundamental idea of finance that the price of an asset such as stock, bond, or real estate investment depends on the present values of future cash flows, and how these present values can be computed.

More on Corporate Finance ePrep Course
Concepts in Finance

##### 5. Cross-Word Puzzle on Biotechnology (Animal Cloning) More on Biotechnology ePrep Course

##### 6. Worked Example on Engineering Economy (Project Evaluation)

Question: A small airline executive charter company needs to borrow \$160,000 to purchase a prototype synthetic vision system for one of its business jets. The SVS is intended to improve the pilots’ situational awareness when visibility is impaired. The local (and only) banker makes this statement: “We can loan you \$160,000 at a very favorable rate of 12% per year for a five-year loan. However, to secure this loan, you must agree to establish a checking account (with no interest) in which the minimum average balance is \$32,000. In addition, your interest payments are due at the end of each year, and the principal will be repaid in a lump-sum amount at the end of year five.” What is the true effective annual interest rate being charged?

Answer. The cash-flow diagram from the banker’s viewpoint appears below. When solving for an unknown interest rate, it is good practice to draw a cashflow diagram prior to writing an equivalence relationship. Notice that P0 = \$160,000 − \$32,000 = \$128,000. Because the bank is requiring the company to open an account worth \$32,000, the bank is only \$128,000 out of pocket. This same principal applies to F5 in that the company only needs to repay \$128,000 since the \$32,000 on deposit can be used to repay the original principal. The interest rate (IRR) that establishes equivalence between positive and negative cash flows can now easily be computed:

P0 = F5(P/F, i ′ %, 5) + A(P/A, i ′ %, 5),

\$128,000 = \$128,000(P/F, i ′ %, 5) + \$19,200(P/A, i ′ %, 5).

If we try i ′ = 15%, we discover that \$128,000 = \$128,000.

Therefore, the true effective interest rate is 15% per year.

More on Engineering Economy ePrep Course

##### 7. Video Lesson on Life Science (Cloning)

This short video discusses the process called nucleus transfer of cloning the first mammal, the sheep Dolly.

More on Life Science Eprep Course

##### 8. Python Programming (Calculating Hourly-Rated Wages)

Code:
def calcWeeklyWages(totalHours, hourlyWage):
if totalHours <= 40:
totalWages = hourlyWage*totalHours
else:
overtime = totalHours – 40
totalWages = hourlyWage*40 + (1.5*hourlyWage)*overtime

hours = float(input(‘Enter hours worked: ‘))
wage = float(input(‘Enter dollars paid per hour: ‘))
total = calcWeeklyWages(hours, wage)
print(‘Wages = \$’,total)

Output:
Enter hours worked: 50
Enter dollars paid per hour: 12
Wages = \$ 660.0

#### 9. Economics (Costs in the Short Run and in the Long Run)

1. The division of total costs into fixed and variable costs will vary from firm to firm

• Some costs are fixed in the short run, but all are variable in the long run.
• For example, in the long run, a firm could choose the size of its factory.
• Once a factory is chosen, the firm must deal with the short-run costs associated with that plant size.

2. The long-run average-total-cost curve lies along the lowest points of the short-run average-total-cost curves because the firm has more flexibility in the long run to deal with changes in production. 3. The long-run average total cost curve is typically U-shaped, but is much flatter than a typical short-run average-total-cost curve.

4. The length of time for a firm to get to the long run will depend on the firm involved.

5. Definition of the short-run: the period of time in which some factors of production cannot be changed.

6. Definition of the long-run: the period of time in which all factors of production can be altered.

#### 10. Discrete Mathematics (Recurrence Relation)

Question: Solution: ### Example Applications of Mathematics in Various Fields

Here are some examples of applications of mathematics:

Applications of Exponential and Logarithmic Functions to Solve Growth Problems

• level of absorption of drugs
• forensic science to determine time of death
• growth of population, tumor, bacteria

Applications of Linear Programming to Solve Optimization Problems

• Social program planning
• investment-asset allocation
• agriculture-crop planning
• nutrition planning

Applications of Functions and Their Graphs

Deb Farace of Pepsico testified that she shared the mathematical model on how sale is impacted by weather with buyers and resulted in increase in sale because the buyers were able to better place buy orders according to demand due to different weather conditions.

Applications of Derivatives

Richard Mizak of Kroll Zollo Cooper testified that he used mathematical models to help distressed companies to improve their operations.