Chapter 1 Functions and Models 1.2 Mathematical Models: A catalog of Essential Functions A mathematical model is a mathematical description of a real world situations such as the size of a population, the demand for a product, the speed of a falling object. Purpose: To understand the situation and make predictions about future behaviour. Real-world problem Formulate Mathematical Solve Mathematical Interpret model conclusions Real-world predictions Test Figure 1. Steps of mathematical modeling Tpes of Mathematical Models: Linear Models: A linear function is of the form = f() = m + b where m is the slope and b is the -intercept. For a linear function, the rate of change is constant. The slope of a horizontal line is. If m = 0, it is called a constant function. Its equation is given b is. The slope of a vertical line is. 1
2 Question: What tpes of slopes correspond to increasing and decreasing lines? Two important line formulas: (1) Slope-intercept form: = m + b where m is the slope and b is the -intercept. (Good for graphing.) (2) Point-slope form: 1 = m( 1 ) where m is the slope and ( 1, 1 ) is a point on the line. (Good for finding am equation of a line) Recall: The slope is given b m = 2 1 2 1, where ( 1, 1 ) and ( 2, 2 ) are two disticnt points on the line. Eample 1. Given two points (1, 1) and (2, 3) on a line, find (1) The slope intercept form. (2) The point-slope form. (3) The change in corresponding to a decrease of 7 in.
Eample 2. The manager of a furniture factor finds that it costs $1800 to manufacture 200 chairs in one da and $2400 to produce 300 chair in one da. 3 (1) Epress the cost as a function of the number of chairs produced, assumed that it is linear. (2) What is the slope and what does it represent? (3) What is the -intercept of the graph and what does it represent?
4 Linear Regression: Goal: Given a table of (, ) data, find a best line that fits the data points. Eample 3. A compan manager wants to establish a relationship between the sales of a certain product and the price. The compan research department provides the following data:. Price () $35 $40 $45 $48 $50 Dail Sales () 80 75 68 66 63 (1) Make a scatter plot of the data. Use our calculators. Enter our and values into lists. To do this, hit STAT and ENTER. If ou have anthing in L1 or in L2, cursor up to the name of the list, hit CLEAR and ENTER. Now enter values in column L1 and values in column L2, one at a time b pressing ENTER after each number. Hit 2nd, Y=, ENTER, ENTER to turn on the scatter plot. To view the scatter plot, hit ZOOM and 9. (2) Find and graph the regression line. To find the equation of the linear regression model, first hit STAT, cursor right to CALC and hit 4. To give our regression equation a name before ou hit ENTER, hit VARS, arrow to Y-Vars, and press 1, and then select a name Y1. At this point in time ou should have the following on our calculator screen LinReg(a+b) Y1. Now press ENTER. This will give ou the regression line, where a is the slope and b is the -intescept. To graph it, hit ZOOM and 9 (3) Use the linear model in part (b) to predict the number of units that would be sold at a price of $60. To find the value of a specific input hit VARS, then move right to Y-VARS, hit ENTER twice. Now write the input in paranthesis. For eample write Y1(60) and hit ENTER to see the value.
5 Polnomials: A function P is called a polnomial if P () = a n n + a n 1 n 1 +... + a 1 + a 0, where n is a nonnegative integer and the numbers a 0, a 1,..., a n are constants called the coefficients. The domain of an polnomial is. If the leading coefficient a n 0 then the degree of the polnomial is n. Eample 4. The degree of the polnomial P () = 3 + + 5 is. It s leading coefficient is. A polnomial of degree is called quadratic function. Its graph is alwas a. A polnomial of degree is called cubic function. Eample 5. = 2 2 2 = 3 2 + 1
6 Power Functions: A function of the form f() = a where a is a constant is called a power function. If a = n, where n is a positive integer, the power function is a polnomial with onl one term. Eample 6. = = 2 = 3 = 4 = 5 = 6 Question: Do ou see a pattern? The shape of f() = n depends on whether n is. Also as n increases, the graph of = n becomes flatter near 0 and steeper when 1.
7 If a = 1 n, where n is a positive integer, the power function f() = 1 n = n is called a root function. Eample 7. Describe the domains. = = 3 If a = 1, f() = 1 = 1 n is called the reciprocal function. Its graph is a hperbola with the coordinate aes as its asmptotes. = 1
8 Rational functions: A rational function f is a ratio of two polnomials where P and Q are polnomials. The domain of a rational function is f() = P () Q(), Eample 8. Find the domain of the function f() = 1 2 2. Algebraic functions: 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 polnomials. Eample 9. (1) An rational function is automaticall an algebraic function. (2) f() = 2 + + 2 (3) A more complicated one g() = 3 32 + + ( 5) 3 1
Trigonometric Functions: Trigonometr and trigonometric functions are reviewed on Referece Page 2 and also in Appendi C. 9 Eample 10. 1 1 f() = cos 2π 3 2 π π π 2 π 2 π 3 2 π 2π 2π 3 2 π π π 2 π 2 π 3 2 π 2π 1 f() = sin 1 (1) What is the domain for both function? (2) What is the range for both function?
10 Eponential function: The eponential functions are of the form f() = a, where the base a is a positive constant. If a > 1, then the eponential function is increasing and if 0 < a < 1, then it is decreasing. Eample 11. = 2 ( 1 = 2 ) (1) What is the domain for both function? (2) What is the range for both function?
Logarithmic function: The logarithmic functions are of the form f() = log a, where 11 the base a is a positive constant. Logarithmic functions are of eponential functions. Eample 12. = log 2 () = log 10 () (1) What is the domain for both function? (2) What is the range for both function?
12 Eample 13. Classif the following functions as one of the tpes of functions that we have discussed. (1) f() = 1 + 2 1 3 (2) g() = 12 (3) h() = 6 (4) p() = 1 2 + 1 2 (5) q() = tan()