Chapter 2 Limits and Continuity PDF

Chapter 2 Limits and Continuity Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 1 of 100  Section 2.1 Rates of Change and Tangents Lines to Curves Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide...

Chapter 2 Limits and Continuity Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 1 of 100  Section 2.1 Rates of Change and Tangents Lines to Curves Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 2 of 100 A moving object’s average speed during an interval of time is found by dividing the distance covered by the time elapsed. The unit of measure is length per unit time: kilometers per hour, meters per second, or whatever is appropriate to the problem at hand. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 3 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 4 of 100 Example 2 Let’s find speed of the falling rock (y = 16𝑡 2 ) at t = 1, and t = 2 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 5 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 6 of 100 Geometrically, the rate of change of 𝑓 𝑜𝑣𝑒𝑟 [𝑥1 , 𝑥2 ] is slope of the secant. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 7 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 9 of 100 Example 3 Find the slope of the tangent line to the parabola 𝑦 = 𝑥 2 at P(2,4). Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 10 of 100 Example 4 Figure shows population 𝑝(𝑡) of fruit flies. Find average growth rate from day 23 to day 45. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 11 of 100 Example 5 How fast was the population in Example 4 growing on day 23? 350−0 On day 23 the population was increasing at a rate of about =16.7 flies /day. 35−14 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 12 of 100 Section 2.2 Limit of a Function and Limit Laws Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 13 of 100 Example 1 How does the given function behave near 𝑥 = 1? Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 14 of 100 Example 1 How does the given function behave near 𝑥 = 1? Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 15 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 16 of 100 Example 2 The limit value of a function does not depend on how the function is defined at the point being approached. Consider the three functions in Figure 2.8. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 17 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 18 of 100 Example 4 A function may not have a limit at a particular point. Some ways that limits can fail to exist are illustrated in Figure 2.10. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 19 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 20 of 100 Evaluating limits. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 21 of 100 Example 7 Evaluate limit of the function at 𝑥 = 1. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 22 of 100 Example 8 Estimate the limit at 𝑥 = 0. True limit value is lim 𝑓(𝑥) = 0.05 𝑥→0 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 23 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 24 of 100 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 25 of 100 The Sandwich Theorem is also called the Squeeze Theorem or the Pinching Theorem. Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 26 of 100 Example 10 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 27 of 100 Example 11 The Sandwich Theorem helps us establish several important limit rules: lim 𝑠𝑖𝑛𝑥 = 0 𝑥→0 lim (1 − 𝑐𝑜𝑠𝑥) = 0 𝑥→0 Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 28 of 100  Section 2.3 The Precise Definition of a Limit Thomas' Calculus, 14e in SI Units Copyright © 2020 Pearson Education Ltd. Slide 29 of 100  yields 𝑥 − 4