AMTH122 Survey of Calculus with Applications

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AMTH122Survey of Calculus with Applications

• Rule of Derivative

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Homework Due today

• Topic 7 Exercise 5, 9,11,17,19,21,31
• Derivative
• Instantaneous Rate of Change

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Derivative

• The derivative of f(x)
• The slope of the tangent line at (x,f(x))
• The instantaneous rate of change of f(x) at the
  point (x,f(x))
• Notation f(x), y, dy/dx, d/dx (f(x))

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Example

• Calculate the derivative of function f(x) 3x-2,
  f(x)
• Estimate f(2) and interpret the meaning.

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Calculate derivative

• Find the derivatives
• f(x)6
• f(x)25x

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Derivative and The graph of the function
F(a)gt0 the graph of function increases
near xa F(a)lt0 the graph of function
decreases near xa
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Non existence of Derivatives

• If f (x) does not exist for some x ? a, then we
  say f(x) is nondifferentiable at x a.
• This happens when the graph of f(x)
• is discontinuous at x a
• has a sharp corner at x a
• has a vertical tangent line at x a

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Simple Power Rule

• If
• Then
• Example
• y x5 y 5 x5-1 5x4
• y x1/2 y ½(x1/2 1) ½(x
  -1/2)
• Know the properties

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Constant Multiplier Rule

• If
• Then
• Example

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Sum and Difference Rule

• If
• Then
• If
• Then

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More examples

• Find the derivative of the each of the following
  function

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Numerical Derivatives using TI-83/84

• The calculator typically gives only approximate
  derivatives -Not exact answer!
• Given f(x) x2, calculate f(2)
• Press MATH, select nDeriv(x2,x,2)
• Or
• Plot graph of x2, press 2nd trace, select dy/dx

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Business Application

• Total Cost C(x)
• Marginal Cost C(x)
• Total Revenue R(x)
• Marginal Revenue R(x)
• Total Profit P(x)
• Marginal Profit P(x)

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Example Cell phone company

• A company that manufactures cell phones find the
  cost of producing x phones is given by
• And the profit is given by
• The company currently produces 100 phones, find
• Marginal cost
• Marginal profit and interpret your answer.
• Marginal revenue

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Before our next class

• Finish Topic 8 Exercise 1, 7, 9,31,33,37
• Read Topic 9
• Do the Warm-Up and Check Your Understanding
  Exercises
• Short Quiz on Wednesday
• Average rate of change
• Definition of instantaneous rate of change, slope
  of curve, derivative
• Derivative