DERIVATIVES

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DERIVATIVES

• CHAPTER 3

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DEFINITION OF DERVIATIVE

• The derivative of f at the point (a,f(a)) equals
  the slope of the tangent to the graph at that
  point.
• Finding the derivative using the definition.

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DERIVATIVE AS THE SLOPE

• Local Linearity
• Equation of the tangent line
• y-f(a) f(a)(x-a)

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RATE OF CHANGE

• Budget deficit as rate of change of the national
  debt
• Velocity as rate of change of position
• Acceleration as rate of change of velocity
• Marginal Cost
• Marginal Profit
• Reaction Rate
• Growth Rate

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Derivative as a function

• Definition of a derivative as a function
• At a point
• As a function

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Derivatives as a function

• Notations for the derivative (differentiation
  operators)
• dot notation sometimes used in physics
• (Newton)

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Derivatives as a function

• Things to notice
• f(x) has a horizontal tangent, f(x) 0
• f(x) is increasing, f(x)gt0
• f(x) is decreasing, f(x)lt0

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Derivatives as a function

• The graph of the derivative function
• Sketching the graph of the derivative function
• from a graph
• from a table
• from an equation

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Derivatives from a Graph Table
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Derivative from a Graph Table
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Differentiation Formulas

• Derivative of a constant
• Derivative of
• Derivative of a constant times a function
• Derivative of the sum and difference of functions
• Derivative of a multiple
• Derivative of a quotient

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Differentiability

• Definition of differentiable
• At a point
• On an interval
• Theorem If f is differentiable at xa, then it
  is continuous at xa.

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Ways a function can fail to be differentiable

• Endpoint
• Corner, kink, or cusp
• Discontinuity
• Vertical tangent

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Examples
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Exercise

• A ball is dropped from the top of the Empire
  State building to the ground below. The height,
  y, of the ball above the ground ( in feet) is
  given as a function fo time, t, (in seconds) by
• Find the velocity of the ball at time t. What
  is the sign of the vleocity? Why is this to be
  expected?
• Show that acceleration of the ball is a constant.
  What are the value and sign of this constant?
• When does the ball hit the ground, and how fast
  is it going at that time? Give your answer in
  ft/sec and in mph.

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Exercise

• The height of a sand dune (in centimeters) is
  represented by ,
  where t is measured in years sine 1995. Find
  f(5) and f(5). Using units, explain what each
  means in terms of the sand dune.

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Exercise

• The period T, of a pendulum is given in terms of
  its length ,l, by
  where g is the
• acceleration due to gravity ( a constant) .
• Find dT/dl.
• What is the sign of dT/dl? What does this tell
  you about the period of pendulums?

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Product Rule
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Rates of Change in the Natural and Social
Sciences

• Review
• Average rate of change slope of secant line
• Instantaneous rate of change slope of tangent
  line

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Rates of Change in the Natural and Social Sciences

• Physics
• Velocity rate position changes
• Acceleration rate velocity changes
• Other rates

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Rates of Change in the Natural and Social Sciences

• Chemistry
• Reaction rate
• Isothermal compressibility

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Rates of Change in the Natural and Social Sciences

• Biology
• Growth rate
• Velocity gradient

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Rates of Change in the Natural and Social Sciences

• Economics
• Marginal cost
• Note that the cost function is a continuous
  approximation to a discrete function
• Marginal profit
• Other Sciences