Limits and Continuity 1 1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.2 THE CONCEPT OF LIMIT 1.3 COMPUTATION OF LIMITS 1.4 CONTINUITY AND ITS CONSEQUENCES 1.5 LIMITS INVOLVING INFINITY; ASYMPTOTES 1.6 FORMAL DEFINITION OF THE LIMIT 1.7 LIMITS AND LOSS OF SIGNIFICANCE ERRORS © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Secant Lines The slope of a straight line is the change in y divided by the change in x. The line through two points on a curve is called a secant line . © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Secant Lines © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Slope at a Point So, what do we mean by the slope of a curve at a point? The answer can be visualized by graphically zooming in on the specified point. Continuing in this way, we obtain successively better estimates of the slope. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.2 Estimating the slope of a curve. Estimate the slope of y = sin x at x = 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.2 Estimating the slope of a curve. A good estimate of the slope of the curve at the point (0, 0) would then appear to be 1. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Arc Length of a Curve A second problem requiring the power of calculus is that of computing distance along a curved path (the curve’s arc length). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Arc Length of a Curve The distance of the curve shown must be greater than . © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE Arc Length of a Curve The approximation improves as the number of line segments increases. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.3 Estimating the Arc Length of a Curve © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

1.1 A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE 1.3 Estimating the Arc Length of a Curve © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12