2.1 Functions and Their Graphs. Copyright Cengage Learning. All rights reserved.

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1 2.1 Functions and Their Graphs Copyright Cengage Learning. All rights reserved.

2 Functions A manufacturer would like to know how his company s profit is related to its production level; a biologist would like to know how the size of the population of a certain culture of bacteria will change over time; a psychologist would like to know the relationship between the learning time of an individual and the length of a vocabulary list; and a chemist would like to know how the initial speed of a chemical reaction is related to the amount of substrate used.

3 Functions In each instance, we are concerned with the same question: How does one quantity depend upon another? The relationship between two quantities is conveniently described in mathematics by using the concept of a function. The set A is called the domain of the function. The set B is called the range of the function. It is customary to denote a function by a letter of the alphabet, such as the letter f.

4 Functions We can think of a function f as a machine. The domain is the set of inputs (raw material) for the machine, the rule describes how the input is to be processed, and the values of the function (range) are the outputs of the machine. A function machine Figure 1

5 Evaluating Functions Example Let the function f be defined by the rule f(x) = 2x 2 x + 1. Find: a. f(1) b. f( 2) c. f(a) d. f(a + h)

6 Applied Example 3 Packaging An open box is to be made from a rectangular piece of cardboard 16 inches long and 10 inches wide by cutting away identical squares (x inches by x inches) from each corner and folding up the resulting flaps (Figure 3). (a) The box is constructed by cutting x by x squares from each corner. (b) The dimensions of the resulting box are (10 2x) by (16 2x) by x. Figure 3

7 Applied Example 3 Packaging Find an expression that gives the volume V of the box as a function of x. What is the domain of the function?

8 Graph Example The graph of a function f is shown in Figure 5. The graph of f Figure 5

9 Graph Example (cont.) a. What is the value of f(3)? The value of f (5)? b. What is the height or depth of the point (3, f(3)) from the x-axis? The point (5, f(5)) from the x-axis? c. What is the domain of f? The range of f?

10 Graph Example 2 Sketch the graph of the function f defined by

11 2.2 The Algebra of Functions Copyright Cengage Learning. All rights reserved.

12 Operations of Functions Let S(t) and R(t) denote the federal government s spending and revenue, respectively, at any time t, measured in billions of dollars. The graphs of these functions for the period between 2004 and 2009 are shown in Figure 11. R(t) S(t) gives the federal budget deficit (surplus) at any time t. Figure 11

13 Operations of Functions The difference function D is difference between R and S or D = R S. It has the same domain as the functions S and R. The graph of the function D is shown in Figure 12. The graph of D(t) Figure 12

14 Operations on Functions Example Let f(x) = and g(x) = 2x + 1. Find the sum s, the difference d, the product p, and the quotient q of the functions f and g.

15 Application of Functions The total cost of operating a business is thus given by the sum of the variable costs and the fixed costs, or C(x) = V(x) + F(x) The total profit of a business is the difference between the total revenue realized and the total cost incurred or P(x) = R(x) C(x)

16 Composition of Functions In general, the composition of a function g with a function f is defined as follows. The function g f (read g circle f ) is also called a composite function.

17 Composition of Functions The interpretation of the function h = g f as a machine is illustrated in Figure 13, and its interpretation as a mapping is shown below. The function h = g f viewed as a mapping Figure 14 The composite function h = g f viewed as a machine Figure 13

18 Composite Example Let f(x) = x 2 1 and g(x) = +1. Find: a. The rule for the composite function g f. b. The rule for the composite function f g.

19 2.4 Limits Copyright Cengage Learning. All rights reserved.

20 Introduction to Calculus Historically, the development of calculus by Isaac Newton ( ) and Gottfried Wilhelm Leibniz ( ) resulted from the investigation of the following problems: 1. Finding the tangent line to a curve at a given point on the curve (Figure 26a) What is the slope of the tangent line T at point P? Figure 26(a)

21 Introduction to Calculus 2. Finding the area of a planar region bounded by an arbitrary curve (Figure 26b) What is the area of the region R? Figure 26(b)

22 Introduction to Calculus The problem of finding the rate of change of one quantity with respect to another is mathematically equivalent to the geometric problem of finding the slope of the tangent line to a curve at a given point on the curve. It is precisely the discovery of the relationship between these two problems that spurred the development of calculus in the seventeenth century and made it such an indispensable tool for solving practical problems.

23 Introduction to Calculus The following are a few examples of such problems: Finding the velocity of an object Finding the rate of change of a bacteria population with respect to time Finding the rate of change of a company s profit with respect to time Finding the rate of change of a travel agency s revenue with respect to the agency s expenditure for advertising

24 A Real-Life Example From data obtained in a test run conducted on a prototype of a maglev (magnetic levitation train), which moves along a straight monorail track, engineers have determined that the position of the maglev (in feet) from the origin at time t (in seconds) is given by s = f(t) = 4t 2 (0 t 30) where f is called the position function of the maglev.

25 A Real-Life Example The position of the maglev at time t = 0, 1, 2, 3,..., 10, measured from its initial position, is f(0) = 0 f(1) = 4 f(2) = 16 f(3) = 36, f(10) = 400 feet. A maglev moving along an elevated monorail track Figure 27

26 A Real-Life Example Suppose we want to find the instantaneous velocity of the maglev at t = 2. This is just the velocity of the maglev as shown on its speedometer at that precise instant of time. Obviously, we can compute the position of the maglev at any time t as we did earlier for some selected values of t. Using these values, we can then compute the average velocity of the maglev over an interval of time.

27 A Real-Life Example For example, the average velocity of the train over the time interval [2, 4] is given by or 24 feet/second.

28 A Real-Life Example Although this is not quite the velocity of the maglev at t = 2, it does provide us with an approximation of its velocity at that time. Can we do better?

29 A Real-Life Example Now, let s describe this process in general terms. Let t > 2. Then, the average velocity of the maglev over the time interval [2, t] is given by By choosing the values of t closer and closer to 2, we obtain a sequence of numbers that give the average velocities of the maglev over smaller and smaller time intervals. This sequence of numbers should approach the instantaneous velocity of the train at t = 2.

30 A Real-Life Example Let s try some sample calculations. Taking the sequence t = 2.5, 2.1, 2.01, 2.001, and , which approaches 2, we find: The average velocity over [2, 2.5] is or 18 feet/second. The average velocity over [2, 2.1] is or 16.4 feet/second. and so forth.

31 A Real-Life Example These results are summarized in Table 1. These computations suggest that the instantaneous velocity of the train at t = 2 is 16 feet/second.

32 Intuitive Definition of a Limit Similarly, if we take a sequence of values of t approaching 2 from the left, such as t = 1.5, 1.9, 1.99, 1.999, and , we obtain the results shown in Table 2. Observe that g(t) approaches the number 16 as t approaches 2 this time from the left-hand side.

33 Intuitive Definition of a Limit In other words, as t approaches 2 from either side of 2, g(t) approaches 16. In this situation, we say that the limit of g(t) as t approaches 2 is 16, written

34 Intuitive Definition of a Limit The graph of the function g, shown in Figure 28, confirms this observation. As t approaches t = 2 from either direction, g(t) approaches y = 16. Figure 28

35 Intuitive Definition of a Limit Observe that the point t = 2 is not in the domain of the function g [for this reason, the point (2, 16) is missing from the graph of g]. This, however, is inconsequential because the value, if any, of g(t) at t = 2 plays no role in computing the limit. This example leads to the following informal definition.

36 Example 2 Let Evaluate

37 Evaluating the Limit of a Function The following properties of limits, enable us to evaluate limits of functions algebraically.

38 Example 4 Use Theorem 1 to evaluate the following limits.

39 Indeterminate Forms Consider which we evaluated earlier by looking at the values of the function for x near x = 2. If we attempt to evaluate this expression by applying Property 5 of limits, we see that both the numerator and denominator of the function approach zero as x approaches 2; that is, we obtain an expression of the form 0/0.

40 Indeterminate Forms In this event, we say that the limit of the quotient as x approaches 2 has the indeterminate form 0/0. As the name suggests, the meaningless expression 0/0 does not provide us with a solution to our problem.

41 Indeterminate Forms One strategy that can be used to solve this type of problem follows.

42 Example 6 Evaluate

43 Limits at Infinity Suppose we are given the function and we want to determine what happens to f(x) as x gets larger and larger. Picking the sequence of numbers 1, 2, 5, 10, 100, and 1000 and computing the corresponding values of f(x), we obtain the following table of values: From the table, we see that as x gets larger and larger, f (x) gets closer and closer to 2.

44 Limits at Infinity The graph of the function f shown in Figure 34 confirms this observation. We call the line y = 2 a horizontal asymptote. In this situation, we say that the limit of the function f(x) as x increases without bound is 2, written The graph of has a horizontal asymptote at y = 2. Figure 34

45 Limits at Infinity In the general case, the following definition for a limit of a function at infinity is applicable.

46 Example 7 Let f and g be the functions Evaluate:

47 Limits at Infinity All the properties of limits listed in Theorem 1 are valid when a is replaced by or. In addition, we have the following property for the limit at infinity.

48 Example 8 Evaluate

49 Homework Section 2.1 Page 57 3,13,15,25,27,29,31,47,49,77 TEC Page 66 1,5,9 Section 2.2 Page 72 19,21,25,27,31,45,47,66 TEC Page 94 1,3,7 Section 2.4 Page 111 3,5,7,17,19,27,29,49,55,57,59,85 TEC Page 117 1,11

50 2.5 One-Sided Limits and Continuity Copyright Cengage Learning. All rights reserved.

51 One-Sided Limits Consider the function f defined by From the graph of f, we see that the function f does not have a limit as x approaches zero because, no matter how close x is to zero, f(x) takes on values that are close to 1 if x is positive and values that are close to 1 if x is negative. The function f does not have a limit as x approaches zero. Figure 37

52 One-Sided Limits If we decided to take values only to the right of 0, we say that the right-hand limit of f as x approaches zero (from the right) is 1, written If we decided to take values only to the left of 0, we say that the left-hand limit of f as x approaches zero (from the left) is 1, written These limits are called one-sided limits.

53 One-Sided Limits

54 One-Sided Limits The connection between one-sided limits and the two-sided limit defined earlier is given by the following theorem. Thus, the two-sided limit exists if and only if the one-sided limits exist and are equal.

55 Example 1 Let a. Show that exists by studying the one-sided limits of f as x approaches x = 0. b. Show that does not exist.

56 Continuous Functions Thus, a function f is continuous at x = a if the limit of f at x = a exists and has the value f(a). Geometrically, f is continuous at x = a if the proximity of x to a implies the proximity of f(x) to f(a). If f is not continuous at x = a, then f is said to be discontinuous at x = a. Also, f is continuous on an interval if f is continuous at every number in the interval.

57 Continuous Functions The easy way to check graphically if a function is continuous is to check whether the graph can be sketched without lifting one s pencil from the paper. The graph of f is continuous on the interval (a, b). figure 40

58 Example 2 Find the values of x for which each function is continuous. The graph of each function is shown in Figure 41. Figure 41(a) Figure 41(b) Figure 41(c)

59 Example 2 cont d Figure 41(d) Figure 41(e)

60 Properties of Continuous Functions Using these properties of continuous functions, we can prove the following results.

61 Example 3 Find the values of x for which each function is continuous.

62 2.6 The Derivative Copyright Cengage Learning. All rights reserved.

63 An Intuitive Example Think of the blue curve as representing a stretch of roller coaster track. T is parallel to the line of sight. Figure 49 (b)

64 An Intuitive Example When the car is at the point P on the curve, a passenger sitting erect in the car and looking straight ahead will have a line of sight that is parallel to the line T, the tangent to the curve at P. As Figure 49a suggests, the steepness of the curve that is, the rate at which y is increasing or T is the tangent line to the curve at P. decreasing with respect to x is given by the slope of the tangent Figure 49 (a) line to the graph of f at the point P (x, f(x)).

65 Applied Example 1 Social Security Beneficiaries The graph of the function y = N(t), shown in Figure 50,gives the number of Social Security beneficiaries from the beginning of 1990 (t = 0) through the year 2045 (t = 55). The number of Social Security beneficiaries from 1990 through We can use the slope of the tangent line at the indicated points to estimate the rate at which the number of Social Security beneficiaries will be changing. Figure 50

66 Applied Example 1 Social Security Beneficiaries Use the graph of y = N(t) to estimate the rate at which the number of Social Security beneficiaries was growing at the beginning of the year 2000 (t = 10). How fast will the number be growing at the beginning of 2025 (t = 35)?

67 Slope of a Tangent Line To define the tangent line to a curve C at a point P on the curve, fix P and let Q be any point on C distinct from P (Figure 51). The straight line passing through P and Q is called a secant line. As Q approaches P along the curve C, the secant lines approach the tangent line T. Figure 51

68 Slope of a Tangent Line We can describe the process more precisely as follows. Suppose the curve C is the graph of a function f defined by y = f(x). Then the point P is described by P(x, f(x)) and the point Q by Q(x + h, f(x + h)), where h is some appropriate nonzero number (Figure 52a). The points P(x, f(x)) and Q(x + h, f(x + h) Figure 52 (a)

69 Slope of a Tangent Line Observe that we can make Q approach P along the curve C by letting h approach zero. As h approaches zero, Q approaches P. Figure 52 (b)

70 Slope of a Tangent Line Next, using the formula for the slope of a line, we can write the slope of the secant line passing through P(x, f(x)) and Q(x + h, f (x + h)) as As we observed earlier, Q approaches P, and therefore the secant line through P and Q approaches the tangent line T as h approaches zero.

71 Slope of a Tangent Line Consequently, we might expect that the slope of the secant line would approach the slope of the tangent line T as h approaches zero. This leads to the following definition.

72 Rates of Change Thus, we now can find the average rate of change between two points as well as the instantaneous rate of change at one point (which is the slope of the tangent line at that point).

73 The Derivative The limit to find the instantaneous rate of change of f at x, is given a special name: the derivative of f at x.

74 The Derivative Other notations for the derivative of f include:

75 The Derivative The calculation of the derivative of f is facilitated by using the following four-step process.

76 Example 4 Let 2 f ( x) = x 4x a. Find f (x). b. Find the point on the graph of f the tangent line to the curve is horizontal. c. Sketch the graph of f and the tangent line to the curve at the point in part (b). d. What is the rate of change at that point?

77 Example 5 Let f(x) = a. Find f (x). b. Find the slope of the tangent line T to the graph of f at the point where x = 1. c. Find an equation of the tangent line T in part (b).

78 Applied Example 7 Demand for Tires The management of Titan Tire Company has determined that the weekly demand function of their Super titan tires is given by p = f( x) = 144 x a. Find the average rate of change in unit price of a tire if the quantity demanded is between 5000 and 6000 tires, between 5000 and 5100 tires, and between 5000 and 5010 tires. b. What is the instantaneous rate of change of the unit price when the quantity demanded is 5000 units? 2

79 Differentiability and Continuity In practical applications we can encounter continuous functions that fail to be differentiable that is, do not have a derivative at certain values in the domain of the function f. It can be shown that a continuous function f fails to be differentiable at x = a when the graph of f makes an abrupt change of direction at (a, f(a)). We call such a point a corner. A function also fails to be differentiable at a point where the tangent line is vertical, since the slope of a vertical line is undefined.

80 Differentiability and Continuity These cases are illustrated below. The graph makes an abrupt change of direction at x = a. Figure 59(a) The slope at x a is undefined. Figure 59(b)

81 Applied Example 8 Wages Mary works at the B&O department store, where, on a weekday, she is paid $8 an hour for the first 8 hours and $12 an hour for overtime. The function f(x) = gives Mary s earnings on a weekday in which she worked x hours. Sketch the graph of the function f, and explain why it is not differentiable at x = 8.

82 Applied Example 8 Solution The graph of f is shown in Figure 60. Observe that the graph of f has a corner at x = 8 and consequently is not differentiable at x = 8. The function f is not differentiable at (8, 64). Figure 60

83 Differentiability and Continuity In general the continuity of a function at x = a does not necessarily imply the differentiability of the function at that number. The converse, however, is true: If a function f is differentiable at x = a, then it is continuous there.

84 Homework Section 2.5 Page 126 1,3,23,27,33,35,45,51,53,64,65 TEC Page 132 1,3,9 Section 2.6 Page 146 7,15,11,19,23,25,37