1.2 Mathematical Models: A Catalog of Essential Functions

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1 1.2 Mathematical Models: A Catalog of Essential Functions A is a mathematical description of a real-world phenomenon. e.g., the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, the cost of emission reductions The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. The Modeling Process: Given a real-world problem, Use the mathematical model to Use the mathematical conclusions to Use the real-world prediction to a mathematical model. mathematical conclusions. real-world predictions. the real-world problem. A mathematical model is never a completely accurate representation of a physical situation. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. Common Functions Used in Mathematical Modeling: : f(x) = mx + b : f(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 : f(x) = x a : f(x) = P (x) Q(x) : constructed using algebraic operations : f(x) = a x : f(x) = log a x Linear Functions y is a linear function of x means that the graph of the function is a line. We can use the slope-intercept form of the equation of a line to write a formula for the function as where m is the and b is the. A characteristic feature of linear functions: Note: The slope of the graph can be interpreted as the of y with respect to x. 1

2 Example 1. The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. b) What is the slope of the graph and what does it represent? c) What is the y-intercept of the graph and what does it represent? If there is no physical law or principle to help formulate a model, construct an, which is based entirely on collected data. 2

3 Example 2. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures Temperature ( F) Chirping rate (chirps/min) a) Draw a scatter plot of the data. b) Find a model for the data. c) A computer or graphing calculator finds the line by the method of which is to minimize the sum of the squares of the vertical distance between the data points and the line. This gives a better linear model called. The model obtained by the least squares method is C = 4.86T Use this model to estimate the chirping rate when the temperature is 63 F. d) According to this model, at what temperature will the chirping rate be 265 chirps/min? 3

4 Interpolation: Estimation of a value between observed values Extrapolation Prediction of a value outside the region of observations Polynomials A function P is called a if where n is a nonnegative integer and the numbers a 0, a 1, a 2,..., a n are constants called the of the polynomial. The domain of any polynomial is. If the leading coefficient a n 0, then the of the polynomial is n. A polynomial of degree 1 is of the form A polynomial of degree 2 is of the form. and so it is a and is called a. The graph is a : A polynomial of degree 3 is of the form. A graph: and is called a Polynomials are commonly used to model various quantities that occur in the natural and social sciences. Example 3. A ball is dropped from the top of LEX LUTHOR: Drop of Doom, 415 feet above the ground, and its height h above the ground t seconds after being dropped is given by h(t) = t 2. When will the ball hit the ground? 4

5 Power Functions A function of the form There are three common cases: (i) a = n, where n is a positive integer (ii) a = 1/n, where n is a positive integer (iii) a = 1, where a is a constant, is called a In case (i), f(x) is a polynomial with only one term. The following are the graphs of the power functions for n = 1, 2, 3, 4, and 5. If n is even, then f(x) = x n is an even function and its graph is similar to the parabola. If n is odd, then f(x) = x n is an odd function and its graph is similar to that of the cubic function. Notice, however, that as n increases, the graph of y = x n becomes flatter near 0 and steeper when x 1. In case (ii), the function f(x) = x 1/n = n x is a. For n = 2, f(x) = x, whose domain is. 5

6 For n = 3, f(x) = 3 x, whose domain is. For other even values of n, the graph of y = n x is similar to that of y = x. The graph of y = n x for n odd is similar to that of y = 3 x. In case (iii), the function f(x) = x 1 = 1/x is the domain is. whose Its graph is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle s Law. In addition to these three common cases, it is also possible for n to be any real number. Power functions are also used to model species-area relationships, illumination as a function of distance from a light source, and the period of revolution of a planet as a function of its distance from the sun. 6

7 Example 4. The number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S = 0.7A 0.3. a) The cave called Misión Imposible near Puebla, Mexico, has a surface area of A = 60 m 2. How many species of bats would you expect to find in that cave? b) If you discover that four species of bats live in a cave, estimate the area of the cave. Rational Functions A f is a ratio of two polynomials: where P and Q are polynomials. The domain consists of all values of x such that. Example 5. Find the domain of the function f(x) = 2x4 x x 2 4 The graph of f(x) is 7

8 Algebraic Functions A function f is called an if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots). [Note that all of the functions we have looked at so far are also algebraic functions.] The graphs of algebraic functions can assume a variety of shapes! Trigonometric Functions In calculus, the convention is that radian measure is ALWAYS used (except when otherwise indicated). The graphs of the sine and cosine functions: The domain for both the sine and cosine functions is. Thus, for all values of x, we have and the range is The zeros of the sine function: Periodic Property of the Sine and Cosine Functions: For all values of x, The periodic nature of these functionss makes them suitable for modeling repetitive phenomena. e.g., tides, vibrating springs, and sound waves 8

9 The tangent function is related to the sine and cosine functions by the equation Graph: The tangent function is undefined whenever cos x = 0:. Its range is. Exponential Functions The where the base a is a positive constant. The domain is. are the functions of the form and the range is Exponential functions are useful for modeling many natural phenomena, such as population growth and radioactive decay. Graphs: 9

10 Logarithmic Functions The, where the base a is a positive constant, are the inverse functions of the exponential functions. The domain is and the range is. Graph: Example 6. Classify the following functions. a) f(x) = (0.2) x b) g(x) = x 4 c) h(x) = 3 + x2 1 + x d) u(t) = 5 + 2t 9t 7 10