1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 78 § 0.2 Some Important Functions

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 78  Linear Equations  Applications of Linear Functions  Piece-Wise Functions  Quadratic Functions  Polynomial Functions  Rational Functions  Power Functions  Absolute Value Function Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 78 Linear Equations EquationExample y = mx + b (This is a linear function) x = a (This is not the graph of a function)

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 78 Linear Equations EquationExample y = b CONTINUED

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 78 Applications of Linear FunctionsEXAMPLE SOLUTION (Enzyme Kinetics) In biochemistry, such as in the study of enzyme kinetics, one encounters a linear function of the form, where K and V are constants. (a) If f (x) = 0.2x + 50, find K and V so that f (x) may be written in the form,. (b) Find the x-intercept and y-intercept of the line in terms of K and V. Explained above. (a) Since the number 50 in the equation f (x) = 0.2x + 50 is in place of the term 1/V (from the original function), we know the following. 50 = 1/V Multiply both sides by V. 50V = 1 Divide both sides by 50. V = 0.02 Now that we know what V is, we can determine K. Since the number 0.2 in the equation f (x) = 0.2x + 50 is in place of K/V (from the original function), we know the following.

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 78 Applications of Linear Functions Explained above. Therefore, in the equation f (x) = 0.2x + 50, K = 0.004 and V = 0.02. 0.2 = K/V Multiply both sides by V. 0.2V = K Replace V with 0.02. 0.2(0.02) = K (b) To find the x-intercept of the original function, replace f (x) with 0. CONTINUED Multiply.0.004 = K This is the original function. Replace f (x) with 0. Solve for x by first subtracting 1/V from both sides.

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 78 Applications of Linear FunctionsCONTINUED Multiply both sides by V/K. Simplify. Therefore, the x-intercept is -1/K. To find the y-intercept of the original function, we recognize that this equation is in the form y = mx + b. Therefore we know that 1/V is the y-intercept.

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 78 Piece-Wise FunctionsEXAMPLE SOLUTION Sketch the graph of the following function. We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3. Notice that for all values of x greater than 3, there is no line.

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 78 Piece-Wise Functions Now we graph the function f (x) = 4 only for those values of x that are greater than 3. Notice that for all values of x less than or equal to 3, there is no line. CONTINUED

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 78 Piece-Wise Functions Now we graph both functions on the same set of axes. CONTINUED

11. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 78 Quadratic Functions DefinitionExample Quadratic Function: A function of the form where a, b, and c are constants and a 0.

12. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 78 Polynomial Functions DefinitionExample Polynomial Function: A function of the form where n is a nonnegative integer and a 0, a 1,...a n are given numbers.

13. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 13 of 78 Rational Functions DefinitionExample Rational Function: A function expressed as the quotient of two polynomials.

14. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 14 of 78 Power Functions DefinitionExample Power Function: A function of the form

15. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 15 of 78 Absolute Value Function DefinitionExample Absolute Value Function: The function defined for all numbers x by such that |x| is understood to be x if x is positive and –x if x is negative