MM - Mathematical Modeling Module Overview

Mathematical Modeling

Introduction

Have you ever wondered how a plane flies or what makes it possible for you to stream audio or video on your phone? Have you compared cell phone plans or the best place to invest your money? What if you were a year older or an inch shorter, or you lived in a different state or country? There are so many little things that affect our lives and virtually all of them fit some type of mathematical model. Equations determine how a plane is built for maximum aerodynamics, the programming for you to stream audio or video on your phone, how much you pay for your phone, and the return on an investment. Cost of living is different in different states and countries. There are equations relating height to health, occupational success, ability to compete in different sports and genetics. Understanding mathematical models helps us understand the world and our place in it.

Mathematical Modeling Essential Questions

How can an appropriate equation be built by looking at a mathematical pattern?
How can prior knowledge of functions be used to build precise and efficient models?
How do the multiple representation of functions aid in building more efficient and more accurate models?
How can technology be employed to help build mathematical models, and how can it be used to assess the appropriateness of a specific model?
How can we derive and apply the formula for the sum of a finite geometric series?
How can both algebraic and geometric models optimize particular important values?
How can systems of equations and inequalities be used to define feasible regions of solutions to solve problems?
What is the purpose of building constraints for a model, including using constraints to define feasible solutions and using domain restrictions when analyzing graphs to ensure validity of a function?
Why is revision necessary in model building?
Why is a deep knowledge of the various types of basic mathematical functions absolutely necessary in order to build models for real-world phenomena?
Why is building functions, including combining and composing functions, important in the process of mathematical modeling?

Key Terms

The following key terms will help you understand the content in this module.

Absolute Value - The absolute value of a number is the distance the number is from zero on the number line.

Asymptotes - An asymptote is a line or curve that approaches a given curve arbitrarily closely. A graph never crosses a vertical asymptote, but it may cross a horizontal or oblique asymptote.

Base (of a Power) - The number or expression used as a factor for repeated multiplication.

Common Logarithm - A logarithm with a base of 10. A common logarithm is the exponent, a, such that 10^a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 10² = 100.

Compound Interest Formula - A method of computing the interest, after a specified time, and adding the interest to the balance of the account. Interest can be computed as little as once a year to as many times as one would like. The formula is where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times compounded per year, and t is the number of years.

Continuous Compound Interest Formula - Interest that is theoretically, computed and added to the balance of an account each instant. The formula is equation image indicator $LaTeX: A=Pe^{rt}$ $A=Pe^{rt}$ , where A is the ending amount, P is the principal or initial amount, r is the annual interest rate, and t is the time in years.

Decay Factor - The base number "b" with a value 0 < b < 1 in a function of the form equation image indicator $LaTeX: A=ab^{x-h}+k$ $A=ab^{x-h}+k$ where 0 < b < 1.

Degree - The exponent of a number or expression.

Degree of a Polynomial - The largest exponent of x which appears in the polynomial.

Domain - The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.

Estimate - A guess about the size, cost, or quantity of something.

Exponential - A number written with an exponent. For example, 6³ is called an exponential expression.

Exponential Function - A function of the form equation image indicator $LaTeX: A=ab^{x-h}+k$ $A=ab^{x-h}+k$ where a, h, and k are real numbers, b > 0, and a and b are not equal to 1.

Exponential Decay Function - A function of the form equation image indicator $LaTeX: A=ab^{x-h}+k$ $A=ab^{x-h}+k$ where 0 < b < 1.

Exponential Growth Function - A function of the form equation image indicator $LaTeX: A=ab^{x-h}+k$ $A=ab^{x-h}+k$ where b > 1.

Factor - When two or more integers are multiplied, each integer is a factor of the product. "To factor" means to write the number or term as a product of its factors.

Function - A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set.

Geometric Sequence - Is a sequence with a constant ratio between successive terms.

Geometric Series - The expression formed by adding the terms of a geometric sequence.

Graph of a Function - The set of all the points on a coordinate plane whose coordinates make the rule of function true.

Growth Factor - The base number "b" with a value b > 1 in a function of the form equation image indicator $LaTeX: A=ab^{x-h}+k,\:where\:b>1$ $A=ab^{x-h}+k,\:where\:b>1$ .

Integer - The set of numbers ...,-3,-2,-1,0,1,2,3,...

Interest - The percent of the money on deposit (the principal) paid to a lender for the use of the principle.

Interval - A regular distance or space between values. The set of points between two numbers.

Natural Base e - Euler's number e with the approximation of 2.718...

Natural Logarithm - A logarithm with a base of e. ln b is the exponent, a, such that equation image indicator LaTeX: e^a=b $e^a=b$ . The natural logarithm of x is written ln x and represents $LaTeX: \log_ex$ $\log_ex$ . For example, ln 8 = 2.0794415...because $LaTeX: e^{2.0794415...}=8$ $e^{2.0794415...}=8$ .

Pattern - A set of numbers or objects that are generated by following a specific rule.

Power - The exponent of a number or expression, which indicates the number of times the number or expression is used as a factor.

Polynomial - An algebraic expression involving variables with nonnegative integer exponents with one or more unlike terms.

Quadratic Function - A function of degree 2 whose graph is a parabola.

Range - The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs. The range is the output in a function or a relation.

Rate - A comparison of two quantities that have different units of measure.

Recursive - A type of sequence in which successive terms are generated by preceding terms in the sequence.

Scatterplot - The graph of a collection of ordered pairs that allows an exploration of the relationship between the points.

Substitute - To replace one element of a mathematical equation or expression with another.

Sum of finite geometric series - The sum S_n, of the first n terms of a geometric sequence is given by equation image indicator $LaTeX: S_n=\frac{a_1-a_1r^n}{1-r}=\frac{a_1\left(1-r^n\right)}{1-r}$ $S_n=\frac{a_1-a_1r^n}{1-r}=\frac{a_1\left(1-r^n\right)}{1-r}$ , where a₁ is the first term and r is the common ratio (r≠1).

Sum of infinite geometric series - The general formula for the sum S of an infinite geometric series a₁, a₂, a₃, a₄,... with common ratio r where |r|<1 is equation image indicator $LaTeX: S_n=\frac{a_1}{1-r}$ $S_n=\frac{a_1}{1-r}$ . If an infinite geometric series has a sum, i.e. if |r| < 1, then the series is called a convergent geometric series. All other geometric (and arithmetic) series are divergent.

Symmetry - A mirror image across a line such as the x axis, y axis or across the origin.

Three-Dimensional Figure - Figures that have length, width, and height.

Two-Dimensional Figure - Figures that have length and width (no height).

Unit - A fixed amount that is used as a standard of measurement.

Variable - A letter or symbol used to represent a number.

x-intercept - The value on the x-axis where a graph crosses the x-axis.

y-intercept - The value on the y-axis where a graph crosses the y-axis.

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