2 LIMITS

SECTION 2.1THE TANGENT AND VELOCITY PROBLEMSGROUP WORK 2: Slope PatternsWhen introducing this activity, it may be best to fill out the first line of the table with your students, or toestimate the slope at x =−1. If a group finishes early, have them try to justify the observations made in thelast part of Problem 2.Answers:1. (a) 0, 0.2, 0.4, 0.6 (b) 11.52. (a) Estimating from the graph gives that the function is increasing for x < −3.2, decreasing for−3.2 < x < 3.2, and increasing for x > 3.2.(b) The slope of the tangent line is positive when the function is increasing, and the slope of the tangentline is negative when the function is decreasing.(c) The slope of the tangent line is zero somewhere between x =−3.2and−3.1, and somewhere betweenx = 3.1and3.2. The graph has a local maximum at the first point and a local minimum at the second.(d) The tangent line approximates the curve worst at the maximum and the minimum. It approximatesbest at x = 0, where the curve is “straightest,” that is, at the point of inflection.TEC GROUP WORK 3: How Fast do You Slope In?In this exercise students use TEC Module 2.1 to explore how quickly the slopes of secant lines approach theslope of the tangent line.Answers:1. (a) −1, − 1 2 , − 1 3(b) 1.43 < b < 2 (c)2 < b < 3.33 (d) The interval is larger to the right of 2.2. 1.24 < b < 1.54, 1.58 < b < 30. The latter is a much larger interval.3. 3.08 < b < 3.19, −1.95 < b < −1.18 4. a = 4, the ranges are the same, a = 0HOMEWORK PROBLEMSCore Exercises: 1, 2, 4, 5, 7Sample Assignment: 1, 2, 4, 5, 7, 9Exercise D A N G1 × × × ×2 × × × ×4 × ×5 ×7 × × ×9 × ×37