Another look at D=RT

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Another look at DRT

• If you travel 240 miles in a car in 4 hours,
  your average velocity during this time is
• This does not mean that the cars speedometer
  was on 60 mph at all times this is only your
  average velocity during this time interval.

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2.6 Constant velocity

• If a cars cruise control was set at 60 mph
  for 4 hours of travel, what would the shape of
  the graph of distance traveled to elapsed time
  be?
• The graph is a straight line and the slope of
  the line is 60.
• So in general, if the graph of distance to
  time is a straight line, at every instant the
  velocity is constantly the same, that is, at
  every instant the velocity is the slope of the
  line.

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2.6 Varying velocity

• Now lets consider the case when the velocity
  of a car is varying over time.
• What is the velocity of
• the car at time t3?
• If the graph were a straight
• line, the answer would be
• the slope of the line as before.
• Lets zoom-in on the graph
• near the point t3.

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2.6 Varying velocity continued

• We see that after magnifying the graph near
  t3, the curve looks like a straight line between
  the inputs
• t2.992 and t3.004.
• It seems reasonable
• to assume that the curve behaves like the
  straight line and that the velocity is constant
  during this time interval. So at each instant
  during this time, which includes t3, the
  velocity is the slope of this line.

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2.6 Varying velocity continued

• Two points on the graph
• are approximately
• (2.992, 26.85) and
• (3.004, 27.05).
• Therefore the
• velocity at t3 is

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2.6 Varying velocity continued

• For the functions we are studying it can be
  proven that the more you zoom-in on the graph of
  the function at a specified input, the curve will
  look more and more like a particular straight
  line a tangent line.
• Putting this all together, we define the
  instantaneous rate of change of a function at a
  specified input xa to be the slope of the
  tangent line at the point ( a, f(a) ).

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2.6 Instantaneous rate of change

• The instantaneous rate of change of a function
  f at the input xa
• slope of tangent line at xa
• derivative of f at xa
• This is read f prime of a

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2.6 Example of Instantaneous Rate of Change

• The distance in feet traveled by a car moving
  along a straight road x seconds after starting
  from rest is given by
• f(x) 2x2, 0lt x lt30
• Use a tangent line to approximate the
  (instantaneous) velocity of the car at x22.
• Solution On a graph
• of the function, draw a tangent
• line at x22. Then find its slope
• from any 2 points on the line.

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2.6 Limit Definition of Instantaneous Rate of
Change

• The instantaneous rate of change of a function
  f at the input xa is defined by

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2.6 When is a function not differentiable, that
is, f (a) does not exist?

• You have learned that if the graph of a function
  is broken at a point, then the function is not
  continuous at the point. That is, the graph of a
  continuous function is unbroken.
• It can be shown that a differentiable function is
  continuous. This means the graph of a
  differentiable function must be unbroken too.
  But there is another requirement. A function is
  not differentiable wherever the graph has a sharp
  turning point, a cusp or a vertical tangent line
  at the point.
• The function whose graph
• is shown here is not
• differentiable at the
• points x -2, x0 and
• x1.

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2.6 When, What, How and By How Much

• A function has output the weight of an infant
  w(t) in lb and input age t in mo.
• Write a sentence to interpret (explain the
  meaning) of the instantaneous rate of change
  w(3)1.5.
• Answer
• A function has output body temperature of a
  patient F(t) in oF and input t, hours after
  taking a fever-reducing drug. In a sentence,
  interpret F(3) -0.25.

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2.6 Average Rate of Change

• The average rate of change of a function f
  from the input x to the input xh, or over the
  interval x, xh, is given by
• In words, this is the change in the outputs
  divided by the change in the inputs.

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2.6 Geometric Interpretation of ARC

• If a straight line goes thru the two points
  (x1,y1) and (x0, y0), then the slope of the line
  is given by
• So the average rate of change of a function from
  input x0 to input x1, is the same as the slope of
  the straight line going thru the points x0 and
  x1.
• This line is called a secant line.

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2.6 Average rate of change of f between 2 inputs
equals slope of a secant line
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2.6 Example of Average Rate of Change

• The distance in feet traveled by a car moving
  along a straight road x seconds after starting
  from rest is given by
• f(x) 2x2, 0lt x lt30
• For each of the following three time
  intervals, calculate the average velocity of the
  car.
• 22, 23, 22, 22.1, 22,22.01