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Calculus III: Section 14.4
Professor Ensley
Ship Math
October 17, 2011
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 1 / 6
Tangent Plane
Tangent Plane
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 2 / 6
Tangent Plane
Tangent Plane
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 2 / 6
Tangent Plane
Tangent Plane
Example (14.4 #1) Find the equation of the plane tangent tof (x , y) = x2y3 at the point (2, 1, 4). Use this to get an approximation forthe value of f (2.01, 1.02) = (2.01)2(1.02)3.
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 2 / 6
Local Linearity
Local Linearity
Example Find the equation of the plane tangent to f (x , y) = 5x + 4y2 atthe point P = (2, 1, 14). (Note that Example 2 in Section 14.4 proves thatthis answer satisfies the definition of “local linearity.”)
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 3 / 6
Local Linearity
First Order Approximation
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 4 / 6
Differentiability
Differentiability
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6
Differentiability
Differentiability
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6
Differentiability
Differentiability
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6
Differentiability
Differentiability
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6
Linear Approximation
Linear Approximation
Example (Section 14.4 #32) The body mass index (BMI) is defined by
the function f (W ,H2) =W
H2, where W and H are weight and height,
respectively. Estimate the change in BMI if a boy with weight 34 kg andheight 1.3m grows to be 1.32m and 36kg.
Example (Section 14.4 #32) Use a linear approximation to estimate the
value of8.01√
(1.99)(2.01).
Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 6 / 6
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