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Rates of Change and Tangent Lines

Rates of Change and Tangent Lines . 2.4. Slope of a Curve. Tangent to a Curve. In calculus, we often want to define the rate at which the value of a function y = f(x) is changing

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Rates of Change and Tangent Lines

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  1. Rates of Change and Tangent Lines 2.4

  2. Slope of a Curve

  3. Tangent to a Curve In calculus, we often want to define the rate at which the value of a function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. The problem with this is that we only have one point and our usual definition of slope requires two points.

  4. Tangent to a Curve • The process becomes: • Start with what can be calculated, namely, the slope of a secant through P and a point Q nearby on the curve. • Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. • Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

  5. Slope of a Curve

  6. Example Find the slope of the parabola y = x2 at the point P(1,1). Write the equation for the tangent to the parabola at this point.

  7. The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

  8. The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

  9. slope at The slope of the curve at the point is: slope

  10. Slope of a Curve at a Point

  11. is called the difference quotient of f at a. The slope of the curve at the point is: If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.

  12. Example Tangent to a Curve

  13. In the previous example, the tangent line could be found using . If you want the normal line, use the negative reciprocal of the slope. (in this case, ) The slope of a curve at a point is the same as the slope of the tangent line at that point. (The normal line is perpendicular.)

  14. Normal Line • The Normal Line to a curve at a point is the line perpendicular to the tangent at the point.

  15. Let a Find the slope at . Example 4:

  16. Let b Where is the slope ? Example 4:

  17. Example Normal to a Curve

  18. Example Average Rates of Change

  19. These are often mixed up by Calculus students! If is the position function: velocity = slope Review: average slope: slope at a point: average velocity: So are these! instantaneous velocity:

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