One-Variable Calculus

Transcription

1 POLI Mathematical and Statistical Foundations Department of Political Science University California, San Diego September 30, 2010

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3 al Relationships Political Science, economics, sociology, as well as biology, chemistry, and physics are frequently concerned with binary relationship between elements of different sets. For example, we may be interested in the dependence of one quantity upon others. (i.e. a manufacturer might want to know how profit varies with production level). Among these relationships, we are interested in those where there is a correspondence between every element in one of the sets and one and only one element in the other set. A relationship of this kind is called a functional relationships or simply a function.

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5 : Definition s,, Definition. A function defines a rule which assigns to each element in a set A a unique element in a set B. In traditional calculus a function is defined as a relationship between two variables. Denote the variables by x and y. f : A B, assigns to each x A a unique element y B. If every value of x is associated with exactly one value of y, then y is said to be a function of x.

6 Domain, Image, and Range The most common way to denote a function is to replace the term y, called the image of the element x and to write y = f(x) (the symbol f(x) is the value of f at x and it is usually read as f of x.). If f is a function from A to B and S A, we say that f is defined on the set S. The largest set on which f is defined is, of course, the set A. We call A the domain of f. The set f(a) is called the range of f.

7 Domain, Image, and Range (cont.) For example, suppose A = {1, 2, 3}, and B = { 2, 4}, we can define f such that,

8 Domain, Image, and Range (cont.) Because a number on the left leads to exactly one number on the right, we can say that the numbers on the right (i.e. the images) are a function of those on the left. The arrows give us a mapping of domain to range. As you can see, more than one number on the left can lead to the same number on the right, but not vice versa

9 Vocabulary of s It is customary to use x for what is called the independent variable, and y for what is called the dependent variable because its value depends on the value of x. Values that can be taken by the independent variable are, thus, called the variable s domain. Values that can be taken by the dependent variable are called the range. Note also that in our example, we defined the relationship between the elements of the sets A and B using arrows, which represent our correspondence rules. We may define a function, thus, by simply establishing a set of rules.

10 Vocabulary of s (cont.) Alternatively, we may specify our rule using a mathematical equation, such as y = x 2. This equation defines a function from R to itself. For each x R there exists a unique y R which satisfies the rule y = x 2. The domain of this function is R. The range of this function is [0, ) (recall that the product of two negative numbers is positive). We can also write this function as f(x) = x 2 : the dependent variable is the square of x.

11 Explicit and Implicit s Many functions are also given by an algebraic formula. For example, y = f(x) = x 2 x + 1. In this form, the expression is called an explicit function of x. The equation may also be expressed as x 2 x y 1 = 0, and in this case we refer to it as an implicit function of x because the explicit form is implied by the equation.

12 Explicit and Implicit s (cont.) So, if f(x) = x 2 x + 1, what is the value of this function when x = 1, x = 0, x = 1 2? We find the value by inserting the designated x value into the formula: f( 1) = ( 1) 2 ( 1) + 1 = = 3 f(0) = (0) 2 (0) + 1 = 1 f ( ) 1 = 2 ( ) ( ) = = 3 4.

13 Evaluating s s,, Our chief interest will be in rules for evaluating functions defined by formulas. If the domain is not specified, it will be understood that the domain is the set of all real numbers for which the formula produces a real value, and for which it makes sense. For example, given the equation y = 1 x, 1 x is defined for all values of x except zero; so the range is all real numbers except zero.

14 Evaluating s (cont.) Consider now the equation y 2 = x. This equation does not define a function from R to itself. Take any value for which x < 0, and you will find that there is no value of y associated with such value of x. Does this function define a function from [0, ) to R? Again, the answer is no. This time is certainly true that there is a correspondence between every element in the set [0, ) and an element in the range. However, the equation does not assigns to each element in the domain a unique element in the range.

15 Evaluating s (cont.) The equation y 2 = x, though, does define a function from [0, ) to [0, ). Given any x [0, ), there is a unique y [0, ) which satisfies y 2 = x. Recall the discussion about roots from last week: we observe that, for each x 0, f(x) = x.

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17 Cartesian Representations It may be an overstatement to say that a picture is worth a thousand words, yet a picture of a function is very useful. This picture is called the graph of the function. Equations with two variables can be represented by curves on the Cartesian plane. We start by constructing coordinate axes: we construct a pair of mutually perpendicular intersecting lines, one horizontal, the other vertical.

18 Cartesian Representations (cont.) The horizontal line is often called the x-axis. Values of the independent variable are usually represented by points along this axis. We call the vertical line the y-axis. Values of the dependent variable are represented by points along this axis. The point where these lines meet, called the origin, represents zero. The scale of the y-axis does not need to be the same as that for the x-axis. In fact, y and x can have different units, such as distance and time.

19 Cartesian Representations (cont.) We can represent one specific pair of values associated by the function in the following way: Let a represent some particular value for the independent variable x, and let b indicate the corresponding value of y = f(x). Thus, b = f(a) We can now draw a line parallel to the y-axis at distance a from the axis, and another line parallel to the x-axis at distance b. The point P at which these two lines intersect is designated by the pair of values (a, b) for x and y, respectively.

20 Cartesian Representations (cont.) The number a is called the x-coordinate of P, and the number b is called the y-coordinate of P. In the designation of a typical point by the notation (a, b) we will always designate the x-coordinate first and the y-coordinate second.

21 : Examples s,, For example, given f(x) = x 1, its graph is the set of points (a, b) in the plane such that y = x 1. Notice that the function is defined: each vertical line cuts the graph in one and only one place.

22 : Examples s,, Similarly, the equation y = x 2 defines a function from R to itself. Again, each vertical line cuts the graph in one and only one place.

23 : Examples s,, Consider now the correspondence rule { 2x 3 if x 4 g(x) = 2 if x 4.

24 : Examples s,, This correspondence rule does not define a function: the vertical line of the equation x = 4 meets the graph in two points: (4, 2), and (4, 5). In words, the number 4 has two images. But, if we eliminate the second image of 4, for example: { 2x 3 if x 4 h(x) = 2 if x > 4. Then, h is a defined function.

25 Plotting s s,, The most direct way to plot the graph of a function y = f(x) is to make a table of reasonably spaced values of x and of the corresponding values of y = f(x). Then each pair of values can be represented by a point. A graph of the function is obtained by connecting the points with a smooth curve. A function whose graph is unbroken (i.e. it can be drawn without lifting the pencil from the paper) is said to be continuous, while one whose graph has a gap or a hole is discontinuous.

26 Discontinuous s For the functions we will encounter in this class, discontinuities may arise in one of the following two ways: A function defined in several pieces will have discontinuities if the graphs of the individual pieces are not connected to each other. A function defined as a quotient will have a discontinuity whenever the denominator is zero.

27 Discontinuous s (cont.) For example, the graph of the function f(x) = x2 +x 2 x 2, looks like this:

28 Plotting s: Symmetry The graphical representation of certain function can be simplified if we take into account its symmetry; the graph for x is:

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30 Common s s,, Constant : it assigns a single fixed number k to every value of the independent variable, x. Hence, f(x) = k. The graph is just a horizontal line.

31 Common s (cont.) Identity : A function f is identical over R if and only if the image of every real number is that same real number. Formally, if x : f(x) = x.

32 Common s (cont.) Absolute Value. This is no other than the graph for x.

33 Linear s s,, A function defined by an equation in the form y = mx + b, where m and b are constants, is called a linear function because its graph is a straight line. This is a graph of a typical linear function.

34 Linear s (cont.) If a line is not vertical, its direction with respect to the coordinate axes in the plane is described by a number called the slope of the line. The slope is measured by marking two points P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) on the line and computing the ratio or, Slope = change in y coordinate change in x coordinate Slope = y 2 y 1 x 2 x 1

35 Linear s (cont.) Here is another way to find the slope of a straight line if its equation is given. If the linear function is in the form y = mx + b, then the slope is given by the expression in the previous slide. Substituting for y, we have Slope = (mx 2 + b) (mx 1 + b) x 2 x 1 = mx 2 mx 1 x 2 x 1 = m(x 2 x 1 ) x 2 x 1 = m. So, for example the slope of y = 7x 5 is simply m = 7.

36 Polynomials s,, If a 0, a 1, a 2,..., a n are all real numbers, then the equation y = a 0 + a 1 x + a 2 x a n x n defines a function from R to itself. Any value of x which is substituted on the right hand side generates a unique corresponding value of y. If a n 0, we call this function a polynomial of degree n. A polynomial of degree 0 is called a constant.

37 Rational s s,, The quotient of two polynomials is called a rational function. Suppose that g and h are polynomial functions. Let S denote the set R with all the values of x for which h(x) = 0 removed. Then, the equation y = g(x) h(x) defines a function from S to R. Such a function is a rational function.

38 Cubic Polynomial s,, Example The function from R to itself defined by the equation y = x 3 3x 2 + 2x is called a polynomial function of degree 3 (or, more loosely, a cubic polynomial ).

39 Rational : Example Example Let S be the set R with 2 and 2 removed. Then, the equation y = x2 +4, (x ±2) defines a function from S to R. x 2 4

40 Algebra of s s,, If S R and f and g are two functions from S to R, then we define the function f + g to be that function from S to R which satisfies (f + g)(x) = f(x) + g(x) (x S). If λ is any real number, we define λf to be the function from S to R which satisfies (λf)(x) = λf(x) (x S). Again, we define the functions fg and f/g by (fg)(x) = f(x) g(x) (x S) and (f/g)(x) = f(x)/g(x) (x S). For the latter definition to make sense, of course, it is essential that g(x) 0 for all x S.

41 Algebra of s: Composition A somewhat less trivial way of combining functions is to employ the operation of composition. Let S and T be subsets of R and suppose that g : S T and f : T R. Then we define the composite function f g : S R by f g(x) = f(g(x)) (x S). Sometimes f g is called a function of a function.

42 Algebra of s: Composition (cont.) Example Let f : R R be defined by f(x)= x2 1 x 2 +1 be defined by g(x)=x 3. Then, f g(x) : R R is given by the formula (x R) and let g : R R f g(x) = f(g(x)) = {g(x)}2 1 {g(x)} = x 6 1 x

43 Inverse s s,, Suppose that A and B are sets and that f is a function from A to B. This means that each element a A has a unique image b = f(a) B. We say that f 1 is the inverse function to f if f 1 is a function from B to A which has the property that x = f 1 (y) if and only if y = f(x).

44 Inverse s (cont.) Not all functions have inverse functions. In fact, it is clear that a function f : A B has an inverse function f 1 : B A if and only if each b B is the image of a unique a A. (Otherwise f 1 could not be a function). A function which has this property is said to be a 1 : 1 correspondence between A and B.

45 Bounded s s,, Let f be defined on S. We say that f is bounded above on S by the upper bound h if and only if, for any x S, f(x) h. This is the same as saying that the set f(s) = {f(x) : x S} is bounded above by h. If f is bounded above on S, then it follows from the continuum property that it has a smallest upper bound (or supremum) on S.

46 Bounded s (cont.) Suppose that k = sup f(x) = sup f(s). x S

47 Bounded s (cont.) It may or may not be true that, for some η S, f(η) = k. If such value of η does exist, we say that k is the maximum of f on the set S and that this maximum is attained at the point η.

48 Bounded s (cont.) Similar remarks apply to lower bounds and minima. If a function f is both bounded above and below on the set S, then we simply say that f is bounded on the set S. From last week s class, it follows that a function f is bounded on a set S if and only if, for some b, it is true that, for any x S, f(x) b.

49 Bounded s (cont.) Example Let f : (0, ) R be defined by f(x) = 1 x (x > 0). This function is unbounded above on (0, 1]. It is, however, bounded below on (0, 1] and attains a minimum of 1 at the point x = 1.

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51 of s s,, The idea of a limit is at the heart of calculus. A derivative, the fundamental concept of differential calculus, is a limit. An integral, the fundamental concept of integral calculus, is a limit. We will first approach the study of the concept of limit in an intuitive rather than a formal way. Next, I will give a precise mathematical definition.

52 of s (cont.) describe what happens to a function f(x) as its variable x approaches a particular number c. To illustrate this concept, suppose you want to know what happens to the function f(x) = x2 +x 2 x 1 as x approaches 1. Note that f(x) is not defined at x = 1. Yet, we can get a feel for the situation by evaluating f(x) using values of x that get closer and closer to 1 from both the left and the right.

53 of s (cont.) The following table summarizes the behavior of f(x) for x near 1: The function values in this table suggest that f(x) approaches the number 3 as x gets closer and closer to 1 from either side.

54 of s (cont.) This behavior may be described by saying the limit of f(x) as x approaches 1 equals 3 and expressed as lim f(x) = 3. x 1 In more general terms, if a function f(x) is defined for values of x about some fixed number c, and if, as x is confined to smaller and smaller intervals about c, the values of f(x) cluster more and more closely about some specific number L, the number L is called the limit of f(x) as x approaches c. It is customary to express this as, lim f(x) = L. x c

55 of s (cont.) Note that the intervals we use lie around the point of interest c, but that the point itself is not included. In fact, f(c) may be entirely different from lim x c f(x). The graph of f(x) = x2 +x 2 x 1 is a line with a hole at (1,3), and the points (x, y) approach this hole as x approaches 1 from either side.

56 Limit: Definition s,, So far we have discussed limits in an intuitive way. Now we are ready for a precise definition of a limit. Definition Limit. Let f(x) be defined in an interval about x = c, but not necessarily at x = c. If there is a number L such that to each positive number ɛ there corresponds a positive number δ such that f(x) L < ɛ provided 0 < x c < δ, we say that L is the limit of f(x) as x approaches c and write lim f(x) = L. x c

57 Properties of s,, If lim x c f(x) and lim x c g(x) exist, then lim [f(x) + g(x)] = lim f(x) + lim g(x) x c x c x c lim [f(x) g(x)] = lim f(x) lim g(x) x c x c x c lim [k f(x)] = k lim x c f(x) for any constant k x c lim [f(x)g(x)] = [ lim f(x)][ lim g(x)] x c x c x c

58 Limit of a constant function For any number c and a constant k and lim k = k x c lim x = c x c That is, the limit of a constant is the constant itself, and the limit of f(x) = x as x approaches c is c.

59 Limit of a constant function (cont.)

60 of other characteristic functions Limit of the identity function: given the function f(x) = x, lim x = c x c Limit of a linear function: given f(x) = α + βx, (β 0). lim (α + βx) = α + βc x c of Polynomials and Rational s: if p(x) and q(x) are polynomials, then and lim p(x) = p(c) x c p(x) lim x c q(x) = p(c) q(c) if q(c) 0.

61 Continuity s,, Earlier on, we defined a continuous function as one whose graph is an unbroken curve with no holes or gaps. Alternatively, continuity may be thought of as the quality of having parts that are in immediate connection with one another. Not all the functions have this property, but those which do have special features that make them extremely important in the development of calculus.

62 Continuity (cont.) s,, We are ready now to define continuity in terms of limits. Definition Continuity. A function f is continuous at c if (a) f(c) is defined (b) lim x c f(x) exists (c) lim x c f(x) = f(c). If f(x) is not continuous at c, it is said to have a discontinuity there.

63 Differentiation Differentiation is a mathematical technique of exceptional power and versatility. It is one of the two central concepts in calculus and has a variety of applications in the social sciences, including the optimization of functions, and the analysis of rates of change. One of the main ideas of differential calculus is the concept of the derivative. The derivative of a function is simply another function that describes the rate at which a dependent variable changes with respect to the rate at which the independent variable changes.

64 Differentiation To introduce such concept it is customary to look at two problems, one from physics and the other one pertaining geometry. The former is the calculation of the instantaneous velocity of a moving object. The latter is to find the exact slope of the tangent to a function s curve at any specified point along the curve. Both problems lead to the same calculation: the limit of a sequence of ratios when the denominator tends to zero.

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66 Average of a Earlier today, we defined a function as a relationship between two variables. It is interesting to see now what happens to one of the variables in response to changes in the other one. Denote, as before, the variables by x and y. If x increases from an initial value x 1 to a new value x 2, the net change in x is x 2 x 1. If the corresponding y values are y 1 and y 2, the net change in y is y 2 y 1.

67 Average of a (cont.) We define the average rate of change of y to be: average rate of change = change in y change in x = y 2 y 1 x 2 x 1. This average provides a useful summary of what happens to y as a result of the change in x.

68 Average of a : Example Say you are in the business of producing widgets, and your weekly costs c depend on the production level x. If costs rise from $48,500 to $57,800 when production is increased from 500 to 650 units, the average rate of change of costs is, change in c change in x = 57, , = = 62 dollars per unit On the average, costs increase by $62 for each additional widget produced when production is raised from 500 to 650 units.

69 : Notation It is customary to use the following notation to describe changes in a variable. If x goes from x 1 to x 2, this change is called an increment in x and is indicated by the symbol x = (final value) (initial value) = x 2 x 1. The symbol x is read as delta x. The corresponding increment in the dependent variable y is y = y 2 y 1, and the formula for the average rate of change can be rewritten as y x.

70 : Notation (cont.) Notice that we always take (final value) minus (initial value) in discussing an increment in some variable and in forming averages. Example If y = f(x) = 2 + x + x 2, wan can find the average rate of change in y as x increases from 0 to 1 in the following way. Clearly x = x 2 x 1 = 1 0 = 1. And, y 1 = f(x 1 ) = 2 + (0) + (0) 2 = 2; y 2 = f(x 2 ) = 2 + (1) + (1) 2 = 4. Therefore, the corresponding increment in y is y = y 2 y 1 = 4 2 = 2, and the average rate of change is y x = 2 1 = 2.

71 : Example Example A demographic study shows that the number y of residents in a particular city is described by the function f(x) = 100, x 10, where x represents years elapsed since Suppose that we want to find the average rate of change in y (the average population growth rate) over the period from 1965 to 1975: The years 1965 and 1975 corresponds to x values x 1 = 15 and x 2 = 25, so the increment in x is x = x 2 x 1 = = 10 years.

72 : Example (cont.) The corresponding y values and increment y are: y 1 = f(x 1 ) = 100, 000( ) = 100, = 100, 000(2 2) = 282, 800, y 2 = f(x 2 ) = 100, 000( ) = 100, = 100, 000(4 2) = 565, 600, y = y 2 y 1 = 565, , 800 = 282, 800, So, the average growth rate over this particular 10-year period is y x = 282, = 28, 280 residents per year.

73 : Example (cont.) We can use this last example to illustrate the graphical interpretation of the average rate of change.

74 : Example (cont.) The initial and final values of x are marked on the horizontal axis. Over them sit the points Q 1 = (x 1, y 1 ) and Q 2 = (x 2, y 2 ) corresponding to the initial and final situations in 1965 and The fact that x = 10 means that x moves 10 units to the right on the horizontal axis. Because y values represent the height of the graph above the x axis, the increment y = 282, 800 is the amount the graph rises as a result of this change in x.

75 : Example (cont.) If we draw a straight line L between Q 1 and Q 2, the average rate of change y x is just the slope of this straight line. This geometric interpretation is true for any function y = f(x). The average rate of change y x is always equal to the slope of the straight line joining the corresponding initial and final points on the graph: average rate = y x = amount graph rises (or falls) horizontal distance covered.

76 at a single moment Average rates of change are important, but in many situations they are not quite appropriate. Suppose you drive 150 miles in 3 hours. Your average speed for the trip is 50 mph. But your speed can be 70 mph some of the time, and 40 mph at other moments. If you try to argue your way out of a speeding ticket, the fact that your average speed for the trip was 50 mph, even if provable, does not interest the police officer. He is concerned with your speed at the moment he spotted your car.

77 at a single moment (cont.) This example indicates the need for a concept of rate of change at a single moment: the time of apprehension in this case. We will proceed now to describe this concept -the instantaneous rate of change of one variable with respect to another.

78 Instantaneous Suppose the relationship between two variables is given by some function y = f(x). If x 1 is a fixed base value of the independent variable x, consider what happens as x changes by an amount x from x 1 to x 2 = x 1 + x. For most functions, the average rates of change y x will be about the same as long as x is small. In fact, as the increment x is made smaller and smaller, the averages tend toward a limit value interpreted as the instantaneous rate of of change of y when x = x 1. Namely, the average rates hover about the instantaneous rate and get closer to it as x gets close to zero.

79 Instantaneous (cont.) Example Let f(x) = x 2, and consider the base value x 1 = 1. We are looking for the instantaneous rate of change. The change in x is x 2 x 1 = x. The corresponding y values and increment y are and y 1 = f(x 1 ) = (1) 2 = 1 y 2 = f(x 2 ) = (1 + x) 2 = 1 + 2( x) + ( x) 2 y = y 2 y 1 = 2( x) + ( x) 2.

80 Instantaneous (cont.) The formula for the average rate of change y x = 2( x) + ( x)2 x simplifies if we cancel x in the numerator and denominator for any non-zero increment: y x = 2 + x

81 Instantaneous (cont.) The following table lists different values for y x smaller: as x becomes Note that when x is small, the averages are all close to 2.

82 Instantaneous (cont.) In fact, from the equation we can see that the averages y x approach the value 2 as the increment in x approaches zero; the constant term 2 stays fixed while the second term becomes smaller and smaller. Therefore, we are led to assign the value 2 as the limit value of the averages y x. This limit value tells us how fast y is changing when x = 1. For any continuous function defined by, say, y = f(x), if we ask At what rate does y change as x changes?, we can find the answer by taking the following limit: y lim x 0 x.

83 Instantaneous (cont.) In the preceding example, the averages y x = 2 + x tend toward 2 as x gets smaller and smaller; we can express this as or y lim x 0 x = 2. lim (2 + x) = 2. x 0

84 Instantaneous (The Delta Process ) Definition Given a function y = f(x) and some base point x, consider the values of f at nearby points x + x for small nonzero increments x. Then compute: (1.) The value of f at the base point, namely y 1 = f(x) (2.) The value of f at the nearby point, namely y 2 = f(x + x) (3.) The change in y: y = y 2 y 1 = f(x + x) f(x) (4.) The average rate of change going from x to x + x: y x = y 2 y 1 x = f(x+ x) f(x) x y (5.) The limit value: lim x 0 x as the increment x 0. We call this limit value the instantaneous rate of change of y at the base point x.

85 Instantaneous : Examples This limit value can often be computed for an arbitrary base point. Example Instantaneous rate of change of f(x) = x 2 at an arbitrary base point x. At base point x and nearby point x + x, the y values are y 1 = x 2 and the increment in y is y 2 = (x + x) 2 = x 2 + 2x( x) + ( x) 2 y = y 2 y 1 = x 2 + 2x( x) + ( x) 2 x 2 = 2x( x) + ( x) 2

86 Instantaneous : Examples (cont.) Therefore the average rate of change is y x = 2x( x) + ( x)2 x = 2x + ( x) for any nonzero increment x. As x gets smaller and smaller, the base point x does not vary, so the first term 2x stays fixed. The second term becomes very small as x approaches zero, so the instantaneous rate of change is given by the limit value y lim x 0 x = lim (2x + x) = 2x x 0 for any base point x. If x = 1 the rate of change takes the value 2(1)=2, as in the previous example.

87 Instantaneous : Examples (cont.) Example Lets calculate the instantaneous rate of change of f(x) = 1 x arbitrary base point x using the delta process. at an First, notice that the function is undefined at x = 0, so it is meaningless to discuss the instantaneous rate of change there. For base points x 0, we must examine the behavior of y x for small nonzero increments x.

88 Instantaneous : Examples (cont.) We know that at the base point x, y = 1 x and that at the nearby point x + x, so that, y = 1 x + x y x = [ 1 x+ x 1 x ] x for all small increments x.

89 Instantaneous : Examples (cont.) The limit value of y x is not at all apparent from this formula. But if we simplify the expression algebraically, some of the x cancel, and it is easier to see what happens as x gets small. y x = 1 [ x 1 x + x 1 ] x = 1 [ x ( x) ] x 2 + x( x) = x [ x 1 ] x 2 + x( x) 1 = x 2 + x( x) = 1 [ x (x + x) ] x x(x + x)

90 Instantaneous : Examples (cont.) As x approaches zero, all terms involving x become very small, and the remaining terms stay fixed. With this in mind, notice how the denominator in the above equation behaves: lim (x 2 + x( x)) = x 2. x 0 Because the numerator stays equal to 1 and the denominator approaches the value x 2, the quotient approaches 1 as x approaches zero. x 2 Thus, the instantaneous rate of change of this function is y lim x 0 x = 1 x 2.

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92 Derivative of a The instantaneous rate of change of a function y = f(x) is so important that it is given a special name. It is called the derivative of f(x) and is denoted by the symbol f (x). The derivative measures how fast the value of f(x) is changing at the base point being considered. At some base points, f is changing rapidly; at some others, it changes slowly. Thus, the derivative f (x) depends on the base point x, and so should be thought of as a new function derived from the original function f(x).

93 Derivative of a (cont.) Just like Balzac s character in The Deputy of Arcis, who was talking prose without knowing it, we have already computed the derivatives of some simple functions in today s class using the delta process: If f(x) = x 2, its derivative (instantaneous rate of change) is f (x) = 2x If f(x) = 1 x, its derivative (instantaneous rate of change) is f (x) = 1 x 2

94 Derivative: Definition Definition Given a function f(x), we define the derivative f (x) at base point x to be the limit value of averages f (x) = lim x 0 ( f(x + x) f(x) ) x y = lim x 0 x provided this limit value exists. In that case we say f is differentiable at x.

95 Derivative: Notation Various symbols other than f (x) are usually used for the derivative of y = f(x). In this class we will use the most popular symbols, f (x), df, dy, d (f(x)), Df dx dx dx interchangeably.

96 Differentiation Rules The basic method for calculating derivatives is the delta process explained before, which can be time consuming. I shall apply the delta process once and for all to obtain differentiation rules that allow us to write down at a glance the derivatives of simple functions such as polynomials. Thereafter, I will appeal to these rules rather than to the delta process whenever possible.

97 Differentiation Rules (cont.) The first rule gives the derivative of a simple power of x, f(x) = x r, when r is a fixed constant. Two special cases should be mentioned first: If r = 0, then f(x) = x 0 = 1 (constant function everywhere equal to 1). If r = 1, then f(x) = x 1 = x (identity function). The derivatives of these functions are very easy to find by the delta process (thus, I will not do it here you can try it at home if you are bored): If f(x) = 1, then f (x) = 0 If f(x) = x, then f (x) = 1

98 Differentiation Rules (cont.) We also already know that: If f(x) = x 2, its derivative is f (x) = 2x Lets work out the derivative of y = x 3 at an arbitrary base point: At base point x and nearby point x + x, the y values are y 1 = x 3 y 2 = (x + x) 3 = x 3 + 3x 2 ( x) + 3x( x) 2 + ( x) 3 and the increment in y is y = y 2 y 1 = x 3 + 3x 2 ( x) + 3x( x) 2 + ( x) 3 x 3 = 3x 2 ( x) + 3x( x) 2 + ( x) 3

99 Differentiation Rules (cont.) Therefore the average rate of change is y x = 3x 2 ( x) + 3x( x) 2 + ( x) 3 = 3x 2 + 3x( x) + ( x) 2 x for small increments x. Now let x. get smaller and smaller. The base point x does not vary, so the first term 3x 2 stays fixed. The last two terms involve x and obviously become smaller and smaller as x approaches zero, so that the instantaneous rate of change is given by for any base point x. y lim x 0 x = 3x 2

100 Differentiation Rules (cont.) Now that we have worked out the derivatives of a few powers x r, we can put this information together in the following table: Notice that a regular pattern seems to be emerging. If f(x) = x 4, we might guess (correctly) that f (x) = 4x 3.

101 Differentiation Rules (cont.) More generally, if f(x) = x r (and r is a positive integer), it seems reasonable to conjecture that f (x) = rx r 1, because this formula yields the right result for r = 0, 1, 2, 3, 4. This formula is actually valid for any value of r, whether or not it is a positive integer.

102 Differentiation: Power Rule Power Rule. If f(x) = x r, its derivative is d dx (x r ) = rx r 1 for any fixed exponent r

103 Differentiation Rules: Constants So far we have looked at equations in which as x changes its value, so does y. Example Our next step is to find out what effect on the process of differentiating is caused by the presence of constants, that is, of numbers which don t change when x or y changes its value. Lets calculate the instantaneous rate of change of y = x at an arbitrary base point x.

104 Differentiation Rules: Constants (cont.) Just as before, at base point x and nearby point x + x, the y values are y 1 = x y 2 = (x + x) = x 3 + 3x 2 ( x) + 3x( x) 2 + ( x) and the increment in y is y = y 2 y 1 = x x 2 ( x) + 3x( x) 2 + ( x) 3 (x 3 + 5) = 3x 2 ( x) + 3x( x) 2 + ( x) 3 Therefore the average rate of change is 3x 2 + 3x( x) + ( x) 2

105 Differentiation Rules: Constants (cont.) The last two terms involve x and obviously become smaller and smaller as x approaches zero, so that the instantaneous rate of change is given by for any base point x. y lim x 0 x = 3x 2 So the 5 has quite disappeared. It added nothing to the change in x, and does not enter into the derivative. If we had put 7, or any other number, it would have disappeared. So if we take the letter a, or b, or c to represent any constant, it will simply disappear when we differentiate.

106 Derivative of a Constant Consider the graph of a constant function, y = c: it is a horizontal line, and the slope is zero. Thus, for example if f(x) = 5, then f (x) = 0. Derivative of a Constant. For any constant c, its derivative is d (c) = 0 dx That is, the derivative of a constant is zero.

107 Derivative of a Constant (cont.) Consider now what happens if we have a function y = c x, where c is a constant. We can find the derivative of this function using the delta process: if x is a fixed base point and x a nonzero increment, the average rate of change for y = c x is y(x + x) y( x) = c(x + x) cx = (cx + c x) cx = c x Therefore, dy dx = lim y x 0 x = lim c x x 0 x = c.

108 Derivative of a Constant (cont.) Similarly, say you multiply a function f(x) by a constant c. You would obtain a new function c f(c). For example, if f(x) = x 3 and c = 5, c f(c) = 5x 3. We can find the derivative of this function using the delta process: if x is a fixed base point and x a nonzero increment, the average rate of change for y = c f(x) is y x = c f(x + x) c f(x) x = c f(x + x) f(x) x = c f x

109 Derivative of a Constant (cont.) By definition of the derivative f (x), the averages f x approach f (x) as x gets small. Clearly, c ( f x ) must then approach c f (x), so that dy dx = lim y x 0 x = lim f c x 0 x = c f (x) = c df dx.

110 The Constant Multiple Rule The Constant Multiple Rule. derivative is For any constant c, its d df (cf) = c dx dx This rule expresses the fact that the curve of the function y = cf(x) is c times as steep as the curve y = f(x).

111 The Constant Multiple Rule (cont.) Example Let f(x) = 7x 5. The derivative of this function is f (x) = 35x 4. d dx ( 7x 5 ) = ( 7) d dx (x 5 ) = 7(5x 4 ) = 35x 4.

112 Derivative of Sums Suppose we decide now to combine two functions f(x) and g(x) and form their sum f + g by adding their values for each x: (f + g)(x) = f(x) + g(x) Thus, for example, if f(x) = x 2 and g(x) = x, we obtain f + g = x 2 + x. The next rule states that a sum can be differentiated term by term.

113 The Sum Rule The Sum Rule. The derivative of a sum is the sum of the derivatives d df (f + g) = + dg dx dx dx

114 The Sum Rule (cont.) Example Let y = x 2 + 3x 5. We know that d dx (x 2 ) = 2x, and that d dx (3x 5 ) = 15x 4. According to the sum rule, we can simply add these derivatives to get the derivative of the sum x 2 + 3x 5. Namely, d dx (x 2 + 3x 5 ) = d dx (x 2 ) + d dx (3x 5 ) = 2x + 15x 4.

115 Differentiation Rules: Polynomials By combining the sum rule with the power and constant multiple rules, we can differentiate any polynomial. Example Let f(x) = 3x 3 + 5x + 7. d dx = d dx (3x 3 ) + d dx (5x) + d dx (7) = 9x 2 + 5x = 9x 2 + 5

116 Differentiation Rules: Multiplication Suppose that we want now to differentiate the product y = x 2 (3x + 1). We may be tempted to differentiate the factors x 2 and 3x + 1 separately and then multiply our answers. That is, since f (x 2 ) = 2x and f (3x + 1) = 3, we may conjecture that f (x) = 6x. However this answer is wrong. To see this, we can rewrite the function as y = 3x 3 + x 2 and observe that the derivative is 9x 2 + 2x and not 6x.

117 Product Rule The derivative of a product, thus, is not the product of the individual derivatives. Here is the correct formula for the derivative of a product. The Product Rule. If f(x) and g(x) are differentiable, their product (f g)(x) has derivative d dg df (fg) = f(x) + g(x) dx dx dx

118 Derivative of a Product The derivative of a product is the first factor times the derivative of the second plus the second factor times the derivative of the first. According to the product rule, thus, the derivative of the product in the previous example is: d dx [x 2 (3x + 1)] = x 2 d dx (3x + 1) + (3x + 1) d dx (x 2 ) = x 2 (3) + (3x + 1)(2x) = 9x 2 + 2x which is precisely the result that we obtained before when the product was multiplied out and differentiated as a sum.

119 Derivative of a Quotient Recall that the quotient of two functions f(x) and g(x) is defined by taking the quotient of their values for each x: f f(x) (x) = g g(x). This process yields such functions as 1 x where f(x) = 1 and g(x) = x, and y = x (x 2 +1), where f(x) = x and g(x) = x We also know that the quotient function f(x) g(x) wherever g(x) 0. is defined

120 Quotient Rule The derivative of a quotient is not the quotient of the individual derivatives. As with products, there is a differentiation formula for quotients. The Quotient Rule. If f(x) and g(x) are differentiable, their quotient f(x) gx has derivative d dx ( ) f(x) g(x) = g(x) df dx f(x) dg dx (g(x)) 2 for all x where g(x) 0.

121 Quotient Rule (cont.) The quotient rule is probably the most complicated formula we have seen so far. Here is one way to remember it: low dee high, high dee low, low low. Or, you may start by squaring the denominator, which is the denominator in the original quotient, and then take that as the first term in the numerator. Once you do that, you are half way through. The final step is to think of the product rule, except that it contains a minus sign. So, you just switch the terms. Bingo!

122 Quotient Rule (cont.) Example Let f(x) = 1 and g(x) = x 2. Find d dx ( 1 x 2 ). Here f(x) = 1 and g(x) = x 2, so that df dx for all x 0. d ( 1 ) df g(x) dx dx x 2 = = 0, dg dx f(x) dg dx (g(x)) 2 = (x 2 )(0) (1)(2x) (x 2 ) 2 = 2x x 4 = 2 x 3 = 2x. Thus

123 Quotient Rule (cont.) I just picked a very simple example, but the quotient rule is somewhat cumbersome. So, it is better not to use it unnecessarily. Example Let y = 2 3x 2 x x+1 x. We can rewrite the function as y = 2 3 x x x 1, and then apply the power rule term by term to obtain dy dx = 2 3 ( 2x 3 ) ( 1)x 3 = 4 3 x x 2 = 4 3x x 2

124 of Composite s In many situations, a quantity is given as a function of one variable which, in turn, can be though of a function of a second variable. For example, if f(x) = x 3 and g(x) = 5x + 1, then the composite function y = f(g(x)) = (5x + 1) 3. This function y = (5x + 1) 3 may look too complicated at first hand to tackle directly.

125 of Composite s (cont.) However, there is a very simple way to handle these functions: write some symbol such as u for the expression 5x + 1; then the equation becomes Then, y = u 3. dy du = 3u2, and for the expression u = 5x + 1, du dx = 5.

126 of Composite s (cont.) Now, we just have to calculate that is dy dx = dy du du dx, dy dx = 3u2 (5) = (3)(5x + 1) 2 (5) = 15(5x + 1)

127 of Composite s (cont.) This trick is known as the chain rule. Let the composite function g(h(x)) be the function formed from functions g(u) and h(x) by substituting h(x) for u in the formula for g(u).

128 Chain Rule The Chain Rule. If g(u) and h(x) are differentiable functions, d g(h(x)) = g (h(x))h (x) dx

129 Chain Rule To see that this in nothing more than a restatement of the above example, suppose that y = g(h(x)). Then, y = g(u), where u = h(x) and by the chain rule, dy dx = dy du du dx = g (u)h (x) = g (h(x))h (x).

130 Chain Rule (cont.) Example Let f(x) = x 2 + 3x + 2. We can find the derivative of this function using the chain rule. Think of f(x) as the composite function g(h(x)), where g(u) = u = u 1 2, and h(x) = x 2 + 3x + 2.

131 Chain Rule (cont.) Then, g (u) = u = 1 2 u 1 2, and h (x) = 2x + 3, and, by the chain rule, f (x) = g (h(x))h (x) = 1 2 (x 2 + 3x + 2) 1 2 (2x + 3) = 2x x 2 + 3x + 2