1. FUNCTIONS AND MODELS 1

2. 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS In this section, we will learn about: The purpose of mathematical models. FUNCTIONS AND MODELS

3. A mathematical model is a mathematical description—often by means of a function or an equation—of a real-world phenomenon such as: ◦ Size of a population ◦ Demand for a product ◦ Speed of a falling object ◦ Life expectancy of a person at birth ◦ Cost of emission reductions MATHEMATICAL MODELS

4. The purpose of the model is to understand the phenomenon and, perhaps, to make predictions about future behavior. PURPOSE

5. The figure illustrates the process of mathematical modeling. PROCESS

10. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. ◦ A good model simplifies reality enough to permit mathematical calculations, but is accurate enough to provide valuable conclusions. ◦ It is important to realize the limitations of the model. ◦ In the end, Mother Nature has the final say. MATHEMATICAL MODELS

11. When we say that y is a linear function of x, we mean that the graph of the function is a line. LINEAR MODELS

12. A characteristic feature of linear functions is that they grow at a constant rate. constant rate

13. A function P is called a polynomial if where n is a nonnegative integer and the numbers a 0, a 1, a 2, …, a n are constants called the coefficients of the polynomial. POLYNOMIAL FUNCTIONS

14. A polynomial of degree 1 is of the form So, it is a linear function. DEGREE 1

15. A polynomial of degree 2 is of the form It is called a quadratic function. DEGREE 2

16. DEGREE 3 A polynomial of degree 3 is of the form It is called a cubic function.

17. The figures show the graphs of polynomials of degrees 4 and 5. DEGREES 4 AND 5

18. A function of the form where a is constant, is called a power function. POWER FUNCTIONS

20. The general shape of the graph of power functions depends on whether a is odd or even. POWER FUNCTIONS

21. If a is even, then f(x) = x a is an even function, and its graph is similar to the parabola y = x 2. POWER FUNCTIONS

22. If a is odd, then f(x) = x a is an odd function, and its graph is similar to that of y = x 3. POWER FUNCTIONS

23. A function of the form where n is a positive integer, is called a root function. ROOT FUNCTIONS

24. When n = 2, we have the square root function. Note the domain and range of the function.

25. ROOT FUNCTIONS When n = 3, we have the cube root function. Note how the domain and range are different than when n = 2.

26. RECIPROCAL FUNCTION A function of the form is a reciprocal function.

27. A rational function f is a ratio of two polynomials where P and Q are polynomials. ◦ The domain consists of all values of x such that. RATIONAL FUNCTIONS

28. What is the domain of the function? RATIONAL FUNCTIONS

29. A function f is called an algebraic function if it can be constructed using algebraic operations—such as addition, subtraction, multiplication, division, and taking roots— starting with polynomials. ALGEBRAIC FUNCTIONS

30. Any rational function is automatically an algebraic function. Here are two more examples: ALGEBRAIC FUNCTIONS

31. When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. ◦ The figure illustrates some of the possibilities. ALGEBRAIC FUNCTIONS

32. In calculus, the convention is that radian measure is always used (except when otherwise indicated). TRIGONOMETRIC FUNCTIONS

33. ◦ Thus, the graphs of the sine and cosine functions are as shown in the figure. TRIGONOMETRIC FUNCTIONS

34. The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. TRIGONOMETRIC FUNCTIONS

35. For instance, in Example 4 in Section 1.3, we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function: TRIGONOMETRIC FUNCTIONS

36. The tangent function is related to the sine and cosine functions by the equation Its graph is shown. TRIGONOMETRIC FUNCTIONS

37. The remaining three trigonometric functions—cosecant, secant, and cotangent—are the reciprocals of the sine, cosine, and tangent functions. TRIGONOMETRIC FUNCTIONS

38. The exponential functions are the functions of the form where the base a is a positive constant. EXPONENTIAL FUNCTIONS

39. We will study exponential functions in detail in Section 1.5. ◦ We will see that they are useful for modeling many natural phenomena—such as population growth (if a > 1) and radioactive decay (if a < 1). EXPONENTIAL FUNCTIONS

40. The logarithmic functions where the base a is a positive constant, are the inverse functions of the exponential functions. ◦ We will study them in Section 1.6. LOGARITHMIC FUNCTIONS

41. The figure shows the graphs of four logarithmic functions with various bases. LOGARITHMIC FUNCTIONS

42. Transcendental functions are those that are not algebraic. ◦ The set of transcendental functions includes the trigonometric, inverse trigonometric, exponential, and logarithmic functions. ◦ However, it also includes a vast number of other functions that have never been named. TRANSCENDENTAL FUNCTIONS

43. Classify the following functions as one of the types of functions that we have discussed. a.b. c.d.