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Lecture 5

Lecture 5. Basic Calculus for Economists. Analyzing a Limit. We can examine what occurs at a particular point by the limit. Using the function f ( x ) = 2 x – 1, let’s examine what happens near x = 2 through the following chart:.

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Lecture 5

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  1. Lecture 5 Basic Calculus for Economists

  2. Analyzing a Limit We can examine what occurs at a particular point by the limit. Using the function f (x) = 2x – 1, let’s examine what happens near x = 2 through the following chart: We see that as x approaches 2, f (x) approaches 3.

  3. Limits In limit notation we have 3 Definition: We write 2 or as x c, then f (x) L, if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c (on either side of c).

  4. One-Sided Limits • We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. • We write and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line. • In order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

  5. Limit Properties Let f and g be two functions, and assume that the following two limits exist and are finite: Then • the limit of the sum of the functions is equal to the sum of the limits. • the limit of the difference of the functions is equal to the difference of the limits.

  6. Limit Properties(continued) • the limit of a constant times a function is equal to the constant times the limit of the function. • the limit of the product of the functions is the product of the limits of the functions. • the limit of the quotient of the functions is the quotient of the limits of the functions, provided M 0. • the limit of the nth root of a function is the nth root of the limit of that function.

  7. Examples From these examples we conclude that f any polynomial function r any rational function with a nonzero denominator at x = c

  8. Indeterminate Forms It is important to note that there are restrictions on some of the limit properties. In particular if then finding may present difficulties, since the denominator is 0. If and , then is said to be indeterminate. The term “indeterminate” is used because the limit may or may not exist.

  9. Example This example illustrates some techniques that can be useful for indeterminate forms. Algebraic simplification is often useful when the numerator and denominator are both approaching 0.

  10. Difference Quotients Let f (x) = 3x - 1. Find Solution:

  11. Definition of Continuity A function f is continuous at a pointx = c if 1. 2. f (c) exists 3. A function f is continuous on the open interval(a,b) if it is continuous at each point on the interval. If a function is not continuous, it is discontinuous.

  12. Example 1 f (x) = x – 1 at x = 2. 1. The limit exists! 2. f(2) = 1 3. Therefore this function is continuous at x = 2. 1 2

  13. Example 2 f (x) = (x2 – 9)/(x + 3) at x = -31. The limit exists (reduce the fraction). 2. f (-3) = 0/0 is undefined! 3. The function is not continuous at x = -3. (Graph should have an open circle there.)

  14. Example 3 f (x) = |x|/x at x = 0 and at x = 1. 1. Does not exist! 2. f (0) = 0/0 Undefined! 3. The function is not continuous at x = 0. This function is continuous at x = 1.

  15. Limits in Mathematica

  16. Continuity Properties If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval, except for values of x that make the denominator 0.

  17. Examples of Continuous Functions • A constant function is continuous for all x. • For integer n > 0, f (x) = xn is continuous for all x. • A polynomial function is continuous for all x. • A rational function is continuous for all x, except those values that make the denominator 0. • For n an odd positive integer, is continuous wherever f (x) is continuous. • For n an even positive integer, is continuous wherever f (x) is continuous and nonnegative.

  18. Infinite Limits There are various possibilities under which does not exist. For example, if the one-sided limits are different at x = a, then the limit does not exist. Another situation where a limit may fail to exist involves functions whose values become very large as x approaches a. The special symbol  (infinity) is used to describe this type of behavior.

  19. To illustrate this case, consider the function f (x) = 1/(x-1), which is discontinuous at x = 1. As x approaches 1 from the right, the values of f (x) are positive and become larger and larger. That is, f (x) increases without bound. We write this symbolically as Since  is not a real number, the limit above does not actually exist. We are using the symbol  (infinity)to describe the manner in which the limit fails to exist, and we call this an infinite limit.

  20. Example(continued) As x approaches 1 from the left, the values of f (x) are negative and become larger and larger in absolute value. That is, f (x) decreases through negative values without bound. We write this symbolically as The graph of this function is as shown: Note that does not exist.

  21. Infinite Limits and Vertical Asymptotes Definition: The vertical line x = a is a vertical asymptote for the graph of y = f (x) if f (x)   or f (x)  - as x a+ or x a–. That is, f (x) either increases or decreases without bound as x approaches a from the right or from the left. Note: If any one of the four possibilities is satisfied, this makes x = a a vertical asymptote. Most of the time, the limit will be infinite (+ or -) on both sides, but it does not have to be.

  22. Vertical Asymptotesof Polynomials How do we locate vertical asymptotes? If a function f is continuous at x = a, then Since all of the above limits exist and are finite, f cannot have a vertical asymptote at x = a. In order for f to have a vertical asymptote at x = a, at least one of the limits above must be an infinite limit, and f must be discontinuous atx = a. We know that polynomial functions are continuous for all real numbers, so a polynomial has no vertical asymptotes.

  23. Vertical Asymptotes of Rational Functions Since a rational function is discontinuous only at the zeros of its denominator, a vertical asymptote of a rational function can occur only at a zero of its denominator. The following is a simple procedure for locating the vertical asymptotes of a rational function: If f (x) = n(x)/d(x) is a rational function, d(c) = 0 and n(c)  0, then the line x = c is a vertical asymptote of the graph of f. However, if both d(c) = 0 and n(c) = 0, there may or may not be a vertical asymptote at x = c.

  24. Example Let Describe the behavior of f at each point of discontinuity. Use  and - when appropriate. Identify all vertical asymptotes.

  25. Example(continued) Let Describe the behavior of f at each point of discontinuity. Use  and - when appropriate. Identify all vertical asymptotes. Solution: Let n(x) = x2 + x - 2 and d(x) = x2 - 1. Factoring the denominator, we see that d(x) = x2 - 1 = (x+1)(x-1) has two zeros, x = -1 and x = 1. These are the points of discontinuity of f.

  26. Example(continued) Since d(-1) = 0 and n(-1) = -2  0, the theorem tells us that the line x = -1 is a vertical asymptote. Now we consider the other zero of d(x), x = 1. This time n(1) = 0 and the theorem does not apply. We use algebraic simplification to investigate the behavior of the function at x = 1: Since the limit exists as x approaches 1, f does not have a vertical asymptote at x = 1. The graph of f is shown on the next slide.

  27. Example(continued) Vertical Asymptote Point of discontinuity

  28. Limits at Infinity ofPower Functions We begin our consideration of limits at infinity by considering power functions of the form x p and 1/x p, where p is a positive real number. If p is a positive real number, then x p increases as x increases, and it can be shown that there is no upper bound on the values of x p. We indicate this by writing or

  29. Power Functions (continued) Since the reciprocals of very large numbers are very small numbers, it follows that 1/x p approaches 0 as x increases without bound. We indicate this behavior by writing or This figure illustrates this behavior for f (x) = x2 and g(x) = 1/x2.

  30. Power Functions (continued) In general, if p is a positive real number and k is a nonzero real number, then Note:k and p determine whether the limit at  is  or -. The last limit is only defined if the pth power of a negative number is defined. This means that p has to be an integer, or a rational number with odd denominator.

  31. Limits at Infinity of Polynomial Functions What about limits at infinity for polynomial functions? As x increases without bound in either the positive or the negative direction, the behavior of the polynomial graph will be determined by the behavior of the leading term (the highest degree term). The leading term will either become very large in the positive sense or in the negative sense (assuming that the polynomial has degree at least 1). In the first case the function will approach  and in the second case the function will approach -. In mathematical shorthand, we write this asThis covers all possibilities.

  32. Limits at Infinity andHorizontal Asymptotes A line y = b is a horizontal asymptote for the graph of y = f (x) if f (x) approaches b as either x increases without bound or decreases without bound. Symbolically, y = b is a horizontal asymptote if In the first case, the graph of f will be close to the horizontal line y = b for large (in absolute value) negative x. In the second case, the graph will be close to the horizontal line y = b for large positive x. Note: It is enough if one of these conditions is satisfied, but frequently they both are.

  33. Example This figure shows the graph of a function with two horizontal asymptotes, y = 1 and y = -1.

  34. Horizontal Asymptotes of Rational Functions If then • There are three possible cases for these limits. • If m < n, then The line y = 0 (x axis) is a horizontal asymptote for f (x). • 2. If m = n, then The line y = am/bnis a horizontal asymptote for f (x) . • 3. If m > n, f (x) does not have a horizontal asymptote.

  35. Horizontal Asymptotes of Rational Functions (continued) Notice that in cases 1 and 2 on the previous slide that the limit is the same if x approaches  or -. Thus a rational function can have at most one horizontal asymptote. (See figure). Notice that the numerator and denominator have the same degree in this example, so the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. y = 1.5

  36. Example Find the horizontal asymptotes of each function.

  37. Example Solution Find the horizontal asymptotes of each function. Since the degree of the numerator is less than the degree of the denominator in this example, the horizontal asymptote is y = 0 (the x axis). Since the degree of the numerator is greater than the degree of the denominator in this example, there is no horizontal asymptote.

  38. Summary • An infinite limit is a limit of the form(y goes to infinity). It is the same as a vertical asymptote (as long as a is a finite number). • A limit at infinity is a limit of the form(x goes to infinity). It is the same as a horizontal asymptote (as long as L is a finite number).

  39. The Rate of Change For y = f (x), the average rate of change from x = a to x = a + h is The above expression is also called a difference quotient. See Chiang 6.1. It can be interpreted as the slope of a secant. See the picture on the next slide for illustration.

  40. Visual Interpretation Q slope f (a + h) – f (a) Average rate of change = slope of the secant line through P and Q P h

  41. Example 1 The revenue generated by producing and selling widgets is given by R(x) = x (75 – 3x)for 0  x  20. What is the change in revenue if production changes from 9 to 12?

  42. Example 1 The revenue generated by producing and selling widgets is given by R(x) = x (75 – 3x)for 0  x  20. What is the change in revenue if production changes from 9 to 12? R(12) – R(9) = $468 – $432 = $36. Increasing production from 9 to 12 will increase revenue by $36.

  43. Example 1 (continued) The revenue is R(x) = x (75 – 3x)for 0  x  20. What is the average rate of change in revenue (per unit change in x) if production changes from 9 to 12?

  44. Example 1 (continued) The revenue is R(x) = x (75 – 3x)for 0  x  20. What is the average rate of change in revenue (per unit change in x) if production changes from 9 to 12? To find the average rate of change we divide the change in revenue by the change in production: Thus the average change in revenue is $12 when production is increased from 9 to 12. Like Change in Y over change in X.

  45. The Instantaneous Rate of Change Consider the function y = f (x) only near the point P = (a, f (a)). The difference quotient gives the average rate of change of f over the interval [a, a+h]. If we make h smaller and smaller, in the limit we obtain the instantaneous rate of change of the function at the point P:

  46. Visual Interpretation Q Tangent Slope of tangent =instantaneous rate of change. f (a + h) – f (a) P Let h approach 0 h

  47. Instantaneous Rate of Change Given y = f (x), the instantaneous rate of change at x = a is provided that the limit exists. It can be interpreted as the slope of the tangent at the point (a, f (a)). See illustration on previous slide.

  48. The Derivative For y = f (x), we define the derivative of f at x, denoted f ’ (x), to be if the limit exists. If f ’(a) exists, we call fdifferentiable at a. If f ’(x) exist for each x in the open interval (a, b), then f is said to be differentiable over (a, b).

  49. Interpretations of the Derivative • If f is a function, then f ’ is a new function with the following interpretations: • For each x in the domain of f ’,f ’ (x) is the slope of the line tangent to the graph of f at the point (x, f (x)). • For each x in the domain of f ’,f ’ (x) is the instantaneous rate of change of y = f (x) with respect to x. • If f (x) is the position of a moving object at time x, then v = f ’ (x) is the velocity of the object at that time.

  50. Finding the Derivative To find f ‘ (x), we use a four-step process: Step 1. Find f (x + h) Step 2. Find f (x + h) – f (x) Step 3. Find Step 4. Find

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