Derivatives

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Derivatives

• Chapter 3

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Velocity and Other Rates of Change

• 3.4

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Concepts in 3.4

• Instantaneous Rates of change
• Motion Along a Line
• Sensitivity to Change
• Derivatives in Economics
• and why
• Derivatives give the rates at which things change
  in the world.

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Instantaneous Rates of Change
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Average Vs. Instantaneous Rates of Change

• To find an average rate of changecalculate a
  slope.
• To find an instantaneous rate of changetake a
  derivative.

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Example Instantaneous Rates of Change

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Motion Along a Line
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Displacement

• (Change in distance)
• d2 d1
• d(t2) d(t1)
• s(t2) s(t1)

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Average Velocity---v

• (Change in distance)
• (Change in time)
• d2 d1
• t2-t1

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Instantaneous Velocity
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Speed
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Acceleration
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Relationships between Distance, Velocity, and
Acceleration
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Jerk

• Jerk is the derivative of acceleration.
• A sudden change in acceleration is called jerk.
  When a ride in a car or a bus is jerky, it is
  not that the accelerations involved are
  necessarily large but that the changes in
  acceleration are abrupt.

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Jerk
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Free-fall Constants (Earth)
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Example Finding Velocity

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Sensitivity to Change

• When a small change in x produces a large change
  in the value of a function f(x), we say that the
  function is relatively sensitive to changes in x.
  The derivative f(x) is a measure of this
  sensitivity.

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Example Derivatives in Economics

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Derivatives in Economics

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Economic Derivative Relationships

• Marginal Cost is the derivative of Cost.
• rate of change of cost with respect to
  production.
• Marginal Revenue is the derivative of Revenue
• rate of change of revenue with respect to
  production.

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Geometry Derivative Relationships

• The derivative of area is circumference.
• The derivative of volume is area.

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Direction of Motion

• The velocity tells the direction of motion. If
  the velocity is positive, then the object is
  moving forward. If it is negative, then the
  object is moving backward. Zero velocity should
  be the point of change of direction.

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Min and Max

• When a particle reaches its minimum and/or max,
  the speed for that split second is zero.
  Therefore the velocity should be zero. So, set
  the velocity equation equal to zero and solve.
  This is the time at which the max or min occurs.
  Plug the time into the distance equation to get
  the distance at that time.

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Given Distance Function

• How do you find the velocity and acceleration as
  functions of time?
• Find the first and second derivative of the
  distance function.

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Given Distance Function

• How do you find the average velocity?
• Calculate a slope or change in distance over
  change in time.

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Given Distance Function

• How long did it take the rock to reach the
  highest point?
• Take the derivative of the distance to get
  velocity. Set the velocity to zero and solve to
  find the time at which the rock reaches a max or
  min. Check via graphing for now to double check.

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Given Distance Function

• How high did the rock go?
• Plug the time for the max into the distance
  function to get the height.

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Given Distance Function

• Find the displacement during the first 5 seconds.
• Find the position at time zero, then at 5 secs
  and subtract the results.

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Given Distance Function

• When did the rock reach half of its maximum
  height?
• Take half of the max height and set it equal to
  the distance function and solve for t.

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Given Distance Function

• How long was the rock aloft?
• Set the distance formula to zero and solve. The
  largest value for t will represent the time that
  the rock hit the ground or the time it had been
  aloft.

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Given Profit Function

• Find the marginal profit when x desks are sold.
• The derivative of the profit function will give
  you the marginal profit.

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Given Profit Function

• What is the profit when marginal profit is
  greatest?
• Find the second derivative and set to zero, or
  graph the derivative and trace to the highest
  point.

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Given Profit Function

• What is the maximum possible profit?
• Find the limit as x approaches infinity of the
  function or use the table on the calculator and
  set tblstart to a large number.

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Given Graph of Velocity

• When does the particle move forward? Backward?
  Speed up? Slow down?
• Forwardpositive y values graphed
• Backwardnegative y values
• Speeds up when the velocity is negative and
  decreasing and when it is positive and
  increasing.
• Slows down when the velocity is positive and
  decreasing and negative and increasing.

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Given Graph of Velocity

• When is the particles acceleration positive?
  Negative? Zero?
• Positivevelocity is increasing or slope is
  positive.
• Negativevelocity is decreasing or slope is
  negative.
• Zerovelocity is constant or slope is zero.

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Given Graph of Velocity

• When does the particle move at its greatest
  speed?
• When the absolute value of velocity is maximized.
  

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Given Graph of Velocity

• When does the particle stand still for more than
  an instant?
• When the velocity stays at zero for more than
  just at a point.

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Given the Volume Equation

• At what rate does the volume change with respect
  to the radius when r2 ft?
• In other words, at the instant of 2ft what is the
  slope. Take the derivative of the volume and
  plug in 2 for r.