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

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Section 2.6Tangents, Velocities, and Other Rates
of Change

• Goals
• Use limits to find exact values of
• slopes of tangent lines
• velocities
• other rates of change

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Tangents

• If a curve C has equation y f(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 x ?
  a , and
• compute the slope of the secant line PQ

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Tangents (contd)

• Then we let Q approach P along the curve C
  by letting x approach a .
• If mPQ approaches a number m , then we define
  the tangent t to be the line
• through P
• with slope m .
• This is the content of the following definition

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Tangents (contd)

• On the next two slides we illustrate this
  definition

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Tangents (contd)
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Tangents (contd)
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Example

• Earlier we used graphs and tables to find an
  equation of the tangent line to the parabola y
  x2 at the point P(1, 1) .
• Now we can carry out this calculation with
  precision.
• Here a 1 and f(x) x2 , so the slope is
  given by the limit calculations on the next slide

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Example (contd)
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Example (contd)

• Using the point-slope form of the equation of a
  line, we find that an equation of the tangent
  line at (1, 1) is
• y 1 2(x 1) or y 2x 1
• The next three slides show a zoom in on the
  parabola toward the point (1, 1) .
• Note that the curve becomes almost the same as
  its tangent line as we zoom in

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Example (contd)
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Example (contd)
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Example (contd)
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Another Expression

• There is another expression for the slope of a
  tangent line
• We let h x a
• Then x a h
• Thus the slope of secant line PQ is
• The next slide illustrates the case h gt 0

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Another Expression (contd)
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Another Expression (contd)

• In this way the expression for the slope of the
  tangent line becomes
• As an example, we find an equation of the tangent
  line to the hyperbola y 3/x at the point (3,
  1)

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Example (contd)

• Solution Let f(x) 3/x . Then the slope of the
  tangent at (3, 1) is ? .
• The details are given in the calculation on the
  next slide
• followed by a graph of the hyperbola and the
  tangent line.
• Therefore an equation of the tangent line is y
  1 ?(x 3) , or x 3y 6 0 .

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Example (contd)
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Example (contd)
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Velocity

• Suppose an object moves
• along a straight line, and
• according to an equation of motion s f(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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Velocity (contd)

• In the time interval from t a to t a h
  the change in position is f(a h) f(a) , as
  illustrated on the next slide.
• The average velocity over this time interval
  equals the slope of the secant line PQ (shown
  on the subsequent slide)

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Velocity (contd)
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Velocity (contd)
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Instantaneous Velocity

• Now suppose we compute the average velocities
  over shorter and shorter time intervals a, a
  h .
• That is, we let h approach 0 .
• As in the earlier falling ball example, we define
  the instantaneous velocity v(a) at time t a
  to be the limit of the average velocities

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Instantaneous Velocity (contd)

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

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Example

• Again we consider a ball dropped from a height of
  450 m. Find
• the velocity of the ball after 5 seconds
• how fast the ball is traveling when it hits the
  ground.
• Solution We first use the equation of motion s
  f(t) 4.9t2 to find the velocity v(a) after
  a seconds

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Example (contd)

• The velocity after 5 s is
• v(5) (9.8)(5) 49 m/s .

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Example (contd)

• The ball will hit the ground at the time t1
  with s(t1) 450 , that is, 4.9t12 450 .
• Solving for t1 gives t1 9.6 s .
• The velocity of the ball as it hits the ground is
  therefore
• v(t1) 9.8t1 94 m/s

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Other Rates of Change

• Suppose y is a quantity that depends on another
  quantity x .
• Thus, y is a function of x and we writey
  f(x) .
• If x changes from x1 to x2 , then
• the change in x is ?x x2 x1 , and
• the corresponding change in y is
• f(?x) f(x2) f(x1)

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Other Rates (contd)

• The difference quotient
• is called the average rate of change of y with
  respect to x over the interval x1, x2 .
• It can be interpreted as the slope of the secant
  line PQ in the figure on the next slide

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Other Rates (contd)
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Instantaneous Rate of Change

• By analogy with velocity
• we consider the average rate over smaller and
  smaller intervals
• by letting x2 approach x1
• and therefore letting ?x approach 0 .
• The limit of these average rates of change is
  called the (instantaneous) rate of change of y
  with respect to x at x x1 .

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Instantaneous Rate (contd)

• The instantaneous rate of change can be
  interpreted as the slope of the tangent to the
  curve y f(x) at P(x1 , f(x1))

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Example

• Temperature readings T (in C) were recorded
  every hour starting at midnight on a day in
  Whitefish, Montana.
• The time x is measured in hours from midnight.
• The data are given in the table on the next slide

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Example (contd)
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Example (contd)

• Find the average rate of change of temperature
  with respect to time
• from noon to 3 P.M.
• from noon to 2 P.M.
• from noon to 1 P.M.
• Estimate the instantaneous rate of change at
  noon.

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Solution to (a)

• From noon to 3 P.M. the temperature changes from
  14.3 C to 18.2 C, so
• ?T T(15) T(12) 18.2 14.3 3.9 C
• while the change in time is ?x 3 h .
• Therefore, the average rate of change of
  temperature with respect to time is

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Solution to (a) (contd)

• From noon to 2 P.M. the average rate of change is
• From noon to 1 P.M. the average rate of change is

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Solution to (b)

• We plot the given data on the next slide and
  sketch a smooth curve of the temperature
  function.
• Then we draw the tangent at the point P where
  x 12 .
• By measuring the sides of triangle ABC , we
  estimate the slope of the tangent line to be
  10.3/5.5 1.9 .

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Solution to (b) (contd)
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Solution to (b) (contd)

• Therefore the instantaneous rate of change of
  temperature with respect to time at noon is about
  1.9 C .

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Review

• Use of limits to define
• Tangent lines
• Instantaneous velocities
• Other rates of change
• Homework
• 1, 2, 3, 4, 5, 7, 9, 14, 15, 17, 19, 21, 22, 24,
  27, 28