1. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 78 Applied Calculus

2. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 2 of 78  Functions  Derivatives  Applications of Derivative  Techniques of Differentiation  Logarithmic Functions and Applications  The Definite Integrals  The Trigonometric Functions Course Contents …

3. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 3 of 78 Chapter 0 Functions

4. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 4 of 78  Functions and Their Graphs  Some Important Functions  The Algebra of Functions  Zeros of Functions  The Quadratic Formula and Factoring  Exponents and Power Functions  Functions and Graphs in Applications Chapter Outline

5. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 5 of 78 DefinitionExample Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern Rational & Irrational Numbers

6. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 6 of 78 The Number Line A geometric representation of the real numbers is shown below. The Number Line -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

7. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 7 of 78 Open & Closed Intervals DefinitionExample Infinite Interval: The set of numbers that lie between a given endpoint and the infinity Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves [−1, 4] Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves (−1, 4) -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

8. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 8 of 78 Functions A function f is a rule that assigns to each value of a real variable x exactly one value of another real variable y. The variable x is called the independent variable and the variable y is called the dependent variable. We usually write y = f (x) to express the fact that y is a function of x. Here f (x) is the name of the function.EXAMPLES: y = f (x) xy

9. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 9 of 78 Functions in ApplicationEXAMPLE When a solution of acetylcholine is introduced into the heart muscle of a frog, it diminishes the force with which the muscle contracts. The data from experiments of the biologist A. J. Clark are closely approximated by a function of the form where x is the concentration of acetylcholine (in appropriate units), b is a positive constant that depends on the particular frog, and R(x) is the response of the muscle to the acetylcholine, expressed as a percentage of the maximum possible effect of the drug. (a) Suppose that b = 20. Find the response of the muscle when x = 60. (b) Determine the value of b if R(50) = 60 – that is, if a concentration of x = 50 units produces a R = 60% response.

10. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 10 of 78 Functions in Application Replace b with 20 and x with 60. Therefore, when b = 20 and x = 60, R (x) = 75%. This is the given function. (b) Replace x with 50. Replace R(50) with 60. Multiply both sides by b + 50. Distribute on the left side. Subtract 3000 from both sides. Divide both sides by 60. Therefore, b = 33.3 when R (50) = 60. SOLUTION

11. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 11 of 78 FunctionsEXAMPLE SOLUTION If f (x) = x 2 + 4x + 3, find f (a − 2). This is the given function. Replace each x with a – 2. Evaluate (a – 2) 2 = a 2 – 4a + 4. Remove parentheses and distribute. Combine like terms.

12. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 12 of 78 Domain of a Function DefinitionExample Domain of a Function: The set of acceptable values for the variable x. The domain of the function is

13. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 13 of 78 Domain of a Function DefinitionExample Domain of a Function: The set of acceptable values for the variable x. The domain of the function is

14. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 14 of 78 Domain of a Function DefinitionExample Domain of a Function: The set of acceptable values for the variable x. The domain of the function is

15. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 15 of 78 Graphs of Functions DefinitionExample Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xy- plane.

16. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 16 of 78 The Vertical Line Test DefinitionExample Vertical Line Test: A curve in the xy-plane is the graph of a function if and only if each vertical line cuts or touches the curve at no more than one point. Although the red line intersects the graph no more than once (not at all in this case), there does exist a line (the yellow line) that intersects the graph more than once. Therefore, this is not the graph of a function.

17. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 17 of 78 Graphs of EquationsEXAMPLE SOLUTION Is the point (3, 12) on the graph of the function ? This is the given function. Replace x with 3. Replace f (3) with 12. Simplify. Multiply. false Since replacing x with 3 and f (x) with 12 did not yield a true statement in the original function, we conclude that the point (3, 12) is not on the graph of the function.

18. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 18 of 78 Linear Equations EquationExample y = mx + b (This is a linear function) x = a (This is not the graph of a function)

19. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 19 of 78 Linear Equations EquationExample y = b CONTINUED

20. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 20 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.

21. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 21 of 78 Piece-Wise Functions Now we graph the function f (x) = 2 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

22. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 22 of 78 Piece-Wise Functions Now we graph both functions on the same set of axes. CONTINUED

23. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 23 of 78 Quadratic Functions DefinitionExample Quadratic Function: A function of the form where a, b, and c are constants and a 0.

24. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 24 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.

25. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 25 of 78 Rational Functions DefinitionExample Rational Function: A function expressed as the quotient of two polynomials.

26. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 26 of 78 Power Functions DefinitionExample Power Function: A function of the form

27. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 27 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

28. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 28 of 78 Adding FunctionsEXAMPLE SOLUTION Given and, express f (x) + g(x) as a rational function. Replace f (x) and g(x) with the given functions. f (x) + g(x) = Multiply to get common denominators. Evaluate. Add and simplify the numerator. Evaluate the denominator.

29. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 29 of 78 Subtracting FunctionsEXAMPLE SOLUTION Given and, express f (x) − g(x) as a rational function. Replace f (x) and g(x) with the given functions. f (x) − g(x) = Multiply to get common denominators. Evaluate. Subtract. Simplify the numerator and denominator.

30. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 30 of 78 Multiplying FunctionsEXAMPLE SOLUTION Given and, express f (x)g(x) as a rational function. Replace f (x) and g(x) with the given functions. f (x)g(x) = Multiply the numerators and denominators. Evaluate.

31. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 31 of 78 Dividing FunctionsEXAMPLE SOLUTION Given and, express as a rational function. Replace f (x) and g(x) with the given functions. Rewrite as a product (multiply by reciprocal of denominator). Multiply the numerators and denominators. Evaluate.

32. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 32 of 78 Composition of FunctionsEXAMPLE SOLUTION Table 1 shows a conversion table for men’s hat sizes for three countries. The function converts from British sizes to French sizes, and the function converts from French sizes to U.S. sizes. Determine the function h (x) = f (g (x)) and give its interpretation. This is what we will determine.h (x) = f (g (x)) In the function f, replace each occurrence of x with g (x). Replace g (x) with 8x + 1.

33. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 33 of 78 Composition of Functions Distribute. CONTINUED Multiply. Therefore, h (x) = f (g (x)) = x + 1/8. Now to determine what this function h (x) means, we must recognize that if we plug a number into the function, we may first evaluate that number plugged into the function g (x). Upon evaluating this, we move on and evaluate that result in the function f (x). This is illustrated as follows. g (x) f (x) British French French U.S. h (x)h (x) Therefore, the function h (x) converts a men’s British hat size to a men’s U.S. hat size.

34. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 34 of 78 Composition of FunctionsEXAMPLE SOLUTION Given and, find f (g (x)). Replace x by g(x) in the function f (x) f (g (x)) = Substitute. Multiply the numerators and denominators by x + 2. Simplify.

35. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 35 of 78 Zeros of Functions DefinitionExample Zero of a Function: For a function f (x), all values of x such that f (x) = 0.

36. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 36 of 78 Quadratic Formula DefinitionExample Quadratic Formula: A formula for solving any quadratic equation of the form. The solution is: There is no solution if These are the solutions/zeros of the quadratic function

37. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 37 of 78 Graphs of Intersecting FunctionsEXAMPLE SOLUTION Find the points of intersection of the pair of curves. The graphs of the two equations can be seen to intersect in the following graph. We can use this graph to help us to know whether our final answer is correct.

38. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 38 of 78 Graphs of Intersecting Functions To determine the intersection points, set the equations equal to each other, since they both equal the same thing: y. This is the equation to solve. Now we solve the equation for x using the quadratic formula. CONTINUED Subtract x from both sides. Add 9 to both sides. Use the quadratic formula. Here, a = 1, b = −11, and c =18. Simplify.

39. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 39 of 78 Graphs of Intersecting FunctionsCONTINUED We now find the corresponding y-coordinates for x = 9 and x = 2. We can use either of the original equations. Let’s use y = x – 9. Simplify. Rewrite. Simplify.

40. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 40 of 78 Graphs of Intersecting FunctionsCONTINUED Therefore the solutions are (9, 0) and (2, −7). This seems consistent with the two intersection points on the graph. A zoomed in version of the graph follows.

41. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 41 of 78 FactoringEXAMPLE SOLUTION Factor the following quadratic polynomial. This is the given polynomial. Factor 2x out of each term. Rewrite 3 as Now I can use the factorization pattern: a 2 – b 2 = (a – b)(a + b). Rewrite

42. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 42 of 78 FactoringEXAMPLE SOLUTION Solve the equation for x. This is the given equation. Multiply everything by the LCD: x 2. Distribute. Multiply. Subtract 5x + 6 from both sides. Factor. Set each factor equal to zero. Solve.

43. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 43 of 78 Exponents DefinitionExample

44. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 44 of 78 Exponents DefinitionExample

45. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 45 of 78 Exponents DefinitionExample

46. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 46 of 78 Exponents DefinitionExample

47. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 47 of 78 Exponents DefinitionExample

48. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 48 of 78 Applications of ExponentsEXAMPLE SOLUTION Use the laws of exponents to simplify the algebraic expression. This is the given expression.

49. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 49 of 78 Applications of ExponentsCONTINUED Subtract. Divide.

50. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 50 of 78 Geometric ProblemsEXAMPLE SOLUTION Consider a rectangular corral with two partitions, as shown below. Assign letters to the outside dimensions of the corral. Write an equation expressing the fact that the corral has a total area of 2500 square feet. Write an expression for the amount of fencing needed to construct the corral (including both partitions). First we will assign letters to represent the dimensions of the corral. xxxx y y

51. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 51 of 78 Geometric Problems Now we write an equation expressing the fact that the corral has a total area of 2500 square feet. Since the corral is a rectangle with outside dimensions x and y, the area of the corral is represented by: CONTINUED Now we write an expression for the amount of fencing needed to construct the corral (including both partitions). To determine how much fencing will be needed, we add together the lengths of all the sides of the corral (including the partitions). This is represented by:

52. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 52 of 78 Surface AreaEXAMPLE SOLUTION Assign letters to the dimensions of the geometric box and then determine an expression representing the volume and the surface area of the box. First we assign letters to represent the dimensions of the box. x y z Therefore, an expression that represents the volume is: V = xyz.

53. © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 53 of 78 Surface Area Now we determine an expression for the surface area of the box. Note, the box has 5 sides which we will call Left (L), Right (R), Front (F), Back (B), and Bottom (Bo). We will find the area of each side, one at a time, and then add them all up. L: yz x y z CONTINUED R: yz F: xzB: xz Bo: xy Therefore, an expression that represents the surface area of the box is: S = yz + yz + xz + xz + xy = 2yz + 2xz + xy.