2. 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
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3. 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 .
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Slide 3

4. 1.1
A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE
Secant Lines
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Slide 4

5. 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.
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Slide 5

6. 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.
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Slide 6

7. 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.
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Slide 7

8. 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).
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Slide 8

9. 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
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Slide 9

10. 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.
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Slide 10

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
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Slide 11

12. 1.1
A BRIEF PREVIEW OF CALCULUS: TANGENT LINES AND THE LENGTH OF A CURVE
1.3
Estimating the Arc Length of a Curve
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Slide 12