Tangent Planes and Linear Approximations

Tangent Planes

Suppose a surface S has equation , where has continuous first partial derivatives, and let be a point on S.

Equation of the tangent plane to the surface at

Let and be the curves of intersection of S with the planes and respectively. The direction of the tangent line to the curve

at P is given by The direction of the tangent line to the curve

at P is given by

Normal to the tangent plane:

The tangent plane to S at P is the plane containing the tangent vectors and

Plane through P with normal n: Rearranging terms:

Tangent Plane Example

Find an equation of the tangent plane to the paraboloid at

7 (2,1)( 2) (2,1)( 1) x yz f x f y

( , ) 2 (2,1) 4 x xf x y x f ( , ) 6 (2,1) 6 y yf x y y f

7 4( 2) 6( 1) z x y

Simplifying: 4 6 7 z x y

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The paraboloid and its tangent plane at P:

Linear Approximation

The tangent plane to the graph of at is if and are continuous at P.

The linear function is called the linearization of at

The approximation is called the linear approximation or tangent plane approximation of at

The linear approximation is a good approximation when is near provided that the partial derivatives and exist and are continuous at , that is, provided that the function is differentiable.

Linear Approximation - Example

Consider the function

(a) Explain why the function is differentiable at and find the linearization

Both partial derivatives are continuous at the point, so is differentiable.

2

1 162

( , ) (8,0)

yx x

x ef x y f

2

2

2 132

( , ) (8,0)

y

yy ye

x ef x y f

1 13 ( 8) ( 0)6 3

x y

5( , )6 3 3

yxL x y

(b) Use the linearization to approximate the function at 7.5 0.2 5(7.5,0.2) 2.98336 3 3

L

Compare with the actual value

Differentials

The tangent plane at is an approximation to the function for near

Let be a differentiable function, then the differential of the function at is

∆ 𝑓 ∆ 𝑥 ∆ 𝑦The quantity is an approximation to and it represents the change in height of the tangent plane when changes to

Letting and approach zero, yields the following definition:

Differentials Example

Let (a) Find the differential (b) Use the differential to estimate the change in the function, , when

changes from to

0 .0 5 0.1 x y

( , ) 10 ( , ) 2 x yf x y x f x y y

x ydz f dx f dy

10 2 xdx ydy

(1,2) 10 (1,2) 4 x yf f

(1.05,2.1) (1,2) z f f 2 2 2(5(1.05) 2.1 ) (5(1) 2 ) .9225

(a)

(b) (1,2) (1,2) x yz f x f y

.05( )10 .91)4(0. z

Actual value:

Differentials 3D -Example

Let x, y and z be the dimensions of the box.

The dimensions of a rectangular box are measured to be 70 cm, 55 cm and 30 cm and each measurement is correct to within 0.1 cm.Use differentials to estimate the largest possible error when the surface area of the box is calculated from these measurements.

Differential: ¿ (2 𝑦+2𝑧 )𝑑𝑥+(2 𝑥+2𝑧 )𝑑𝑦+(2 𝑦+2𝑥 )𝑑𝑧

Surface Area:

We are given and To find the largest error in the surface area we use together with , and .