Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32)...

11
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

Transcript of Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32)...

Page 1: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

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

Page 2: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Tangent Plane

Tangent Plane

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 2 / 6

Page 3: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Tangent Plane

Tangent Plane

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 2 / 6

Page 4: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

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

Page 5: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

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

Page 6: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Local Linearity

First Order Approximation

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 4 / 6

Page 7: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Differentiability

Differentiability

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6

Page 8: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Differentiability

Differentiability

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6

Page 9: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Differentiability

Differentiability

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6

Page 10: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

Differentiability

Differentiability

Professor Ensley (Ship Math) Calculus III: Section 14.4 October 17, 2011 5 / 6

Page 11: Calculus III: Section 14webspace.ship.edu/deensley/m213/sec14_4.pdf · Example (Section 14.4 #32) The body mass index (BMI) is de ned by the function f(W;H2) = W H2, where W and H

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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