2'6 Tangents, Velocities, and Other Rates of Change

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2.6 Tangents, Velocities, and Other Rates of
Change

• Tangents If a curve C has equation yf(x) and
  we want to find the tangent line to C at the
  point P(a, f(a)), then we consider a nearby point
  Q(x, f(x)), where and compute the slope
  of the secant line PQ

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2.6 Tangents, Velocities, and Other Rates of
Change

• Then we let Q approach P along the curve C by
  letting x approach a. If approaches a
  number m, then we defined the tangent t to be the
  line through P with slope m.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Definition 1 The tangent line to the curve
  yf(x) at the point P(a, f(a)) is the line
  through P with slope
• provided that this limit exists.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Example1 Find an equation of the tangent line to
  the curve at the given point.

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2.6 Tangents, Velocities, and Other Rates of
Change

• There is another expression for the slope of a
  tangent line that is sometimes easier to use. Let
• hx-a
• Then xah
• So the slope of the secant line PQ is

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2.6 Tangents, Velocities, and Other Rates of
Change

• Notice that as x approaches a, h approaches 0
  (because hx-a) and so the expression for the
  slope of the tangent line in Definition 1 becomes

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2.6 Tangents, Velocities, and Other Rates of
Change

• Example 2 Find an equation of the tangent line
  to the curve at the given point.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Velocities Suppose an object moves along a
  straight line according to an equation of motion
  sf(t), where s is the displacement (directed
  distance) of the object from the origin at time
  t. The function f that describes the motion is
  called the position function of the object.

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2.6 Tangents, Velocities, and Other Rates of
Change

• In the time interval from ta to tah the change
  in position is f(ah)-f(a). The average velocity
  over this time interval is

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2.6 Tangents, Velocities, and Other Rates of
Change

• Now suppose we compute the average velocities
  over shorter and shorter time intervals a, ah.
  In other words, we let h approach 0. We define
  the velocity (or instantaneous velocity) v(a) at
  time ta to be the limit of these average
  velocities

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2.6 Tangents, Velocities, and Other Rates of
Change

• Note This means that the velocity at time ta is
  equal to the slope of the tangent line at P.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Example 3 The displacement (in meters) of a
  particle moving in a straight line is given by
• where t is measured in seconds.
• Find the average velocity over each time
  interval
• (i)3, 4 (ii)3.5, 4 (iii)4, 5
• (iv)4,4.5

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2.6 Tangents, Velocities, and Other Rates of
Change

• (b) Find the instantaneous when t4.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Other Rates of Change Suppose y is a function of
  x and we write yf(x).

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2.6 Tangents, Velocities, and Other Rates of
Change

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2.6 Tangents, Velocities, and Other Rates of
Change

• Example 4 See Example 4 on page 144, and do
  problem 25 of section 2.6.

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2.6 Tangents, Velocities, and Other Rates of
Change

• Homework / Class Exercises (Section 2.6)
• Do problems 5, 7, 12-13, 15-17, 18, 20, 27
  (page 145-147)