Derivatives: Part 4 (Application)

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Derivatives Part 4 (Application)

• Motion Along a Line
• Velocity
• Acceleration
• Jerk
• Applications Economics and Genetics

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Derivatives Applications

• Previously
• To find gradient, m
• To find tangent line equations
• To enable easy sketching of graphs (max/min
  points)
• Today
• Rate of change along a line

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Average and Instantaneous Rate of Change
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Rate of Change Example
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Motion Along a Line
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Velocity (Instantaneous)
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Speed
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Acceleration
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Jerk
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Motion along a line Summary
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Motion along a Line Examples
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Motion along a Line Examples

• 2. The position s in meters, of a particle
  moving along a straight line is given by its
  position function
• s 5t3 - 3t2 t
• where t is in seconds.
• Find the velocity, acceleration and jerk of the
  particle at t 4 seconds.

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Motion along a Line Examples

• 3. The metric fall equation of the ball is
• s 4.9t2 , where s m, t sec.
• (a) How many meters does the ball fall in the
  first 2 seconds.
• (b) What is the velocity, speed, acceleration
  and jerk at t 2 sec?

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Motion along a Line Examples

• 4. A dynamite blast blows a heavy rock straight
  up with a launch velocity of 160 ft/sec. It
  reaches the height of s
  160t-16t2, after t sec.
• (a) How high does the rock go?
• (b) What are the velocity and speed of the rock
  when it is 256 ft above the ground on the way up?
• (c)What is the acceleration of the rock at any
  time, t.

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Application Genetics

• Gregor Meldel worked with peas and discovered
  Medellian Law I and II and was henceforth known
  as Father of Genetics. How he documented-
• y 2p p2
• (pfrequency of smoothdominant 1-pfrequency
  of wrinkledrecessive)
• Via graphical methods of y and y, it is noted
  that a small change in introducing dominant
  alleles into highly recessive population will
  have a drastic effect

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Application Economics

• Marginal costs and Marginal Revenues
• Suppose it costs
• c(x) x3 - 6x2 15x (dollars)
• To produce x radiators when 8 30 radiators are
  produced and that
• r(x) x3 - 3x2 12x
• Gives the dollar revenue from selling x
  radiators. Your shop currently produces 10
  radiators a day. About how much extra will it
  cost to produce one more radiator a day? What is
  your estimated increase in revenue for selling 11
  radiators a day?