Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro
Multivariate Calculus - Bruce E. Shapiro
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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.