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Chapter 2 Limits and The Derivative

Chapter 2 Limits and The Derivative. Section 1 Introduction to Limits. Introduction. As we embark on a journey into the study of calculus, it is reasonable to consider how calculus differs from algebra.

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Chapter 2 Limits and The Derivative

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  1. Chapter 2Limits and The Derivative Section 1 Introduction to Limits

  2. Introduction As we embark on a journey into the study of calculus, it is reasonable to consider how calculus differs from algebra. The words static and dynamic capture the essence of the difference between these two mathematics disciplines. In algebra, we solve equations for a particular value of a variable—a static notion. In calculus, we are interested in how a change in one variable affects another variable—a dynamic notion.

  3. Historical Background Concepts of calculus were developed independently by Isaac Newton (1642–1727) of England and Gottfried Wilhelm von Leibniz (1646–1716) of Germany. They sought methods to solve problems concerning motion. Calculus today is used in the physical sciences, in business, economics, life sciences, and social sciences—any discipline that seeks to understand dynamic phenomena.

  4. Key Concepts of Calculus In this chapter, we introduce the derivative, one of the two key concepts in calculus. In a later chapter, we introduce the integral, the second key concept in calculus. Each of these concepts depend on the notion of a limit, which is explained in Sections 2.1 and 2.2.

  5. Functions and Graphs: Brief Review The graph of a function is the graph of the set of all ordered pairs that satisfy the function. Example: The graph of the function with equation f (x) = 2x – 1 is the set of all ordered pairs (x, f(x)). When x = 2, f(2) = 3 and (x, f(x)) = (2, f(2)) = (2, 3) is a point on the graph of f. For x = 0, f(0) = –1 and (0, f(0)) = (0, –1) is on the graph of f. For x = –1, f(–1) = –3 and (–1, f(–1)) = (–1, –3) is on the graph of f. The domain values 2, 0, and -1 are associated with the x axis and the range values f(2) = 3, f(0) = –1, and f(–1) = –3 are associated with the y axis.

  6. Functions and Graphs: Brief Review (continued) For f (x) = 2x – 1, the graph shows the three points found on the previous slide along with the graph of additional points on the function. For each point, the domain value x is located on the horizontal axis and the function value, f(x) is located on the vertical axis. (2, f(2)) = (2, 4) f(2) = 4 f(x) = 2x – 1 x = –1 x = 0 x = 2 f(0) = -1 f(-1) = -3

  7. Example: Finding Function Values from Its Graph Complete the following table using the graph.

  8. Analyzing a Limit The a limit of a function at some domain value x = a involves examining the behavior of the output values of the function as the input values get increasingly closer to the domain value a. The chart gives function values for the function f (x) = 2x – 1 for domain values near x = 2. Notice that the closer x is to 2, the closer the function f (x) is to 3. We symbolize this limit concept with the notation

  9. Example: Analyzing a Limit Let f(x) = x + 2. Discuss the behavior of the values of f(x) when x is close to 2. Solution: We begin by drawing a graph of f that includes the domain value x = 2. f(x) f(x) = x + 2 x 2

  10. Analyzing a Limit (continued) The thin vertical lines represent values of x that are close to 2. The corresponding horizontal lines identify the value of f(x) associated with each value of x. The graph in the figure indicates that as the values of x get closer to 2 on either side of 2, the corresponding values of f(x) get closer and closer to 4.

  11. Analyzing a Limit (continued) This equation is read as “The limit of f(x) as x approaches 2 is 4.” In this example, the value of the function at 2 and the limit of the function as x approaches 2 are the same.

  12. Definition: Limit

  13. Try this Limit Example With the Absolute Value Function in a Limit

  14. Definition One-Sided Limits If no direction is specified in a limit statement, assume that the limit is two-sided or unrestricted.

  15. Theorem 1 On the Existence of a Limit

  16. Example: Analyze Limits Graphically For x near –1 on either side of –1, the function value f(x) is close to 1. The left- and right-hand limits are equal and agree with the function value.

  17. Example: Analyze Limits Graphically For x near 1 on either side of 1, the function value f(x) is close to 3. The left- and right-hand limits are equal. f(1) is not defined.

  18. Example: Analyze Limits Graphically The abrupt break in the graph at x = 2 indicates that the left- and right-hand limits are not equal. The function exists at x = 2 but the limit does not exist at x = 2.

  19. Theorem 2 Properties of Limits Let f and g be two functions, and assume that Where L and M are real numbers (both limits exist). Then

  20. Theorem 2 Properties of Limits(continued) Let f and g be two functions, and assume that Where L and M are real numbers (both limits exist). Then Each property in Theorem 2 is valid for left and right limits.

  21. Examples: Use Theorem 2 to Evaluate Limits

  22. Theorem 3 Limits of Polynomialand Rational Functions

  23. Examples: Evaluate Limits Find each limit.

  24. Examples: Evaluate Limits Find each limit.

  25. Examples: Evaluate Limits (D) Because the definition of the function does not assign a value to f for x = 2, f(2) does not exist.

  26. Definition: Indeterminant Form The term indeterminate is used because the limit of an indeterminate form may or may not exist. Note that the expression 0/0 does not represent a real number and is never used as a value of a limit. As in the following examples, if a limit has an indeterminate form, further investigation is required to determine if the limit exists and to find its value if it does exist.

  27. Example: Indeterminate Form Find the value of the limit if it exists. In this example, both numerator and denominator approach zero as x approaches 2. This limit is an indeterminate form. Algebraic simplification is used to reduce the fraction by dividing the common factor x – 2 (which happens to be the factor in the numerator and denominator that approaches zero when x approaches 2.) Algebraic simplification is often useful when the numerator and denominator are both approaching 0.

  28. Theorem 4 Limit of a Quotient

  29. Example Determine if the limit is a 0/0 indeterminate form. Find the limit or explain why the limit does not exist.

  30. Example Determine if the limit is a 0/0 indeterminate form. Find the limit or explain why the limit does not exist.

  31. Examples: Determine if the limit is a 0/0 indeterminate form. Find the limit or explain why the limit does not exist. By Theorem 4, this limit does not exist.

  32. Limits of Difference Quotients

  33. Example: Limit of a Difference Quotient-You Try It!

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