21.04.2015 Views

Multivariate Calculus - Bruce E. Shapiro

Multivariate Calculus - Bruce E. Shapiro

Multivariate Calculus - Bruce E. Shapiro

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

LECTURE 13. TANGENT PLANES 95<br />

Solution. From equation 13.2, we calculate that<br />

z − z 0 = (x − x 0 )f x (x 0 , y 0 , z 0 + (y − y 0 )f y (x 0 , y 0 , z 0 )<br />

z − 1 = (x − 1)<br />

(e −2y∣ )<br />

∣<br />

(1,0,1)<br />

+ y<br />

(−2xe −2y∣ )<br />

∣<br />

(1,0,1)<br />

= (x − 1)e 0 + y(−2(1)(e 0 ))<br />

= x − 1 − 2y<br />

Solving for z gives z = x − 2y as the equation of the tangent plane.<br />

<br />

Example 13.3 Find the equation of the tangent plane to the surface<br />

at the point (1, 1, 2).<br />

z = √ x + y 1/3<br />

Solution. Differentiating,<br />

∂f<br />

∂x∣ = 1<br />

(1,1,2)<br />

2 √ x∣ = 1<br />

(1,1,2)<br />

2<br />

∂f<br />

∂y ∣ = 1 ∣<br />

∣∣∣(1,1,2)<br />

(1,1,2) 3y 2/3 = 1 3<br />

From equation 13.2, the tangent plane is therefore<br />

z = z 0 + (x − x 0 )f x (x 0 , y 0 , z 0 ) + (y − y 0 )f y (x 0 , y 0 , z 0 )<br />

= 2 + x − 1 + y − 1<br />

(<br />

2 3<br />

= 2 − 1 2 − 1 )<br />

+ x 3 2 + y 3<br />

= 7 6 + x 2 + y 3<br />

Multiplying through by 7 gives 6z = 7 + 3x + 2y.<br />

<br />

Example 13.4 Find a point on the surface<br />

z = 2x 2 + 3y 2 (13.3)<br />

where the tangent plane is parallel to the plane<br />

8x − 3y − z = 0<br />

.<br />

Math 250, Fall 2006 Revised December 6, 2006.

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!