Calculus I

1
Calculus I

• Section 3.11
• Linear Approximations

2
Homework

• Read (pp 262-267)
• Do (pp 267)1, 2, 3, 4, 5, 7, 9, 13, 17, 22, 23, 37

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Linear Approximation

• For x near a
• gives the linearization of f at a.

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Example

• Atmospheric pressure P decreases as altitude h
  increases. At a temp. of 15C, the pressure is
  101.3 kPa at sea level. 87.1 kPa at h 1 km. and
  74.9 kPa at h 2 km. Use linear approximation to
  estimate the atmospheric pressure at an altitude
  of 3 km.

5
Example
6
Example

• For what values of x will the linear
  approximation be within 0.1 o the true value of
  f(x)?

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Differentials

• The idea of behind linear approx. are used to
  formulate the notation and terminology of
  differentials. If y f(x) where f is a
  differentiable function, then the differential dx
  is an independent variable that is, dx can be
  given the value of any real number. The
  differential dy is then defined in terms of dx by
  the equation

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Another Way

• Recall

9
Example

• Find the differential dy for
• then evaluate dy for x 0 and dx 0.1