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CHAPTER 2

Tangents, Velocities and Other Rates of Change. CHAPTER 2. 2.4 Continuity. Definition The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope lim h - > 0 (f(a+h)-f(a)) / h. animation.

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CHAPTER 2

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  1. Tangents, Velocities and Other Rates of Change CHAPTER 2 2.4 Continuity DefinitionThe tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope lim h -> 0 (f(a+h)-f(a)) / h. animation

  2. Example Find the slope of the tangent line to the curve y = x3at the point (-1,1) by using the definition.

  3. Velocities We define the velocity (or instantaneous velocity) v(a) at time t = a to be the limit of these average velocities: v(a) = lim h --> 0 ( f(a + h) –f(a)) / h This means that the velocity at time t = a is equal to the slope of the tangent line at P.

  4. Example If an arrow is shot upward on the moon with a velocity of 58 m/s, its height (in meters) after t seconds is given by H = 58t – 0.83t2, find the velocity of the arrow after 1 second.

  5. Example Consider the slope of the curve below at each of the given points. The curve represents a displacement function s(t). Estimate the average and instantaneous velocities at times and times intervals given by the instructor.

  6. Other Rates of Changes Instantaneous rate of change = lim x  0y/x = lim x1 x2 ( f(x2) – f(x1)) / (x2 – x1)

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