8/13/2019 Lec4[1]PlanoTangente y AproxLineal

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

MATH 209Calculus, III

Volker Runde

University of Alberta

Edmonton, Fall 2011

8/13/2019 Lec4[1]PlanoTangente y AproxLineal

2/13

MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

The tangent plane problem, I

TaskPut a tangent plane to the surface given by z=f(x, y) atP(x0, y0, z0).

Approach

The tangent plane Phas the form

A(x x0) +B(y y0) +C(z z0) = 0.

Suppose that C= 0, then

z z0 = A

C

=a

(x x0)B

C

=b

(y y0)

=a(x x0

) +b(y y0

).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

The tangent plane problem, II

Approach (continued)

Intersect the tangent plane with the plane y=y0:

z z0 =a(x x0).

Hence: a is the slope of the tangent line at P(x0, y0, z0) toz=f(x, y) in the direction of the x-axis, i.e.,

a=fx(x0,

y0).

Similarly:b=fy(x0, y0).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

The tangent plane problem, III

Theorem

Suppose that f has continuous partial derivatives. Then thetangent plane to the surface z=f(x, y) at P(x0, y0, z0) isgiven by

z z0 =fx(x0, y0)(x x0) +fy(x0, y0)(y y0).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Examples, I

ExampleFind the tangent plane to z=x2 + 3y2 at P(2,1, 7).We have:

z

x = 2x and

z

y = 6y,

so that

z

x(2,1) = 4 and

z

y(2,1) = 6.

We thus have the tangent plane equation:

z 7 = 4(x 2) 6(y+ 1) = 4x 8 6y 6,

i.e.,

z= 4x 6y 7.

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Linearization

Idea

Use the tangent plane to approximate f.In the previous example, set

L(x, y) = 4x 6y 7.

Then L(x, y) f(x, y) if (x, y) (2,1).

Definition

Thelinearizationoff at (a, b) is the function

L(x, y) =f(a, b) +fx(a, b)(x a) +fy(a, b)(y b).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Differentiability, I

Notation

Set:

x= increment ofx,

y= increment ofy,z=f(a+ x, b+ y) f(a, b).

Definition

Ifz=f(x, y), then f isdifferentiableat (a, b) if

z=fx(a, b)x+fy(a, b)y+1x+2y

with 1 0 and 2 0 as (x, y) (0, 0).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Differentiability, II

Idea

f is differentiable at (a, b) if and only ifL, i.e., the tangentplane, approximates f very well at (a, b).

Theorem

If fx and f y arecontinuousat(a, b), then f is differentiable at

(a, b).

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

The differential of a function

Definition

Thedifferentialofz=f(x, y) is defined to be

dz=fx(x, y)dx+fy(x, y)dy

with dx and dybeing independent variables.

Idea

Ifdx= x=x a and dy= y=y b, linearapproximation becomes

f(x, y) f(a, b) +dz.

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Examples, II

Example

Letz=f(x, y) =x2 + 3xy y2,

so that

z

x = 2x+ 3y and

z

y = 3x 2y.

and thus dz= (2x+ 3y)dx+ (3x 2y)dy.

Suppose that xchanges from 2 to 2.05 and ychanges from 3to 2.96.

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Examples, III

Example (continued)

Set x= 2, y= 3, dx= x= 0.05, and dy= y= 0.04.Then

dz= (4 + 9)0.0.5 + (6 6)(0.04) = 13(0.05) = 0.65

andz=f(2.05, 2.96) f(2, 3) = 0.6449.

Idea

z dz, but dz is easier to compute.

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Examples, IV

Example

Let Vbe the volume of a circular cone with height h and base

radius r, i.e.,V =

3r2h,

so that

dV =

V

r dr+

V

hdh=

2

3rh dr+

3 r2

dh.

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MATH 209

Calculus,III

Volker Runde

Tangent

planes

Linearization

and differen-tiability

Differentials

Examples, V

Example (continued)

Suppose that r= 10 cm andh= 25 cm, so thatV 2618 cm3.

Suppose that there is an error in measurement of 0.1 cm, i.e.,dr=dh= 0.1.Then

|dV| 500

3

|0.1| +100

3

|0.1| = 20 63,

i.e., the maximum error in V is 63cm3.