s11wk07su

s11wk07su - Math 23 B Dodson Week 3b Homework 14.4 tangent...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 23 B. Dodson Week 3b Homework: 14.4 tangent plane, diﬀerentials 14.5 chain rule Problem 14.4.3: Find the tangent plane to the surface the z = f (x, y ) = 4 − x2 − 2y 2 at (1, −1, 1). Solution: We recall that the tangent plane at (a, b, f (a, b)) is the plane with equation z − f (a, b) = fx (a, b)(x − a) + fy (a, b)(y − b). 1 We compute fx = 1 (4 − x2 − 2y 2 )− 2 (−2x), 2 1 1 and fy = 2 (4 − x2 − 2y 2 )− 2 (−4y ), 1 so fx (1, −1) = −(4 − 1 − 2)− 2 = −1, 1 fy (1, −1) = 2(4 − 1 − 2)− 2 = 2, giving z − 1 = −(x − 1) + 2(y + 1), or −x + 2y − z + 2 = 0, the plane through (1, −1, 1) with normal < fx , fy , −1 >=< −1, 2, −1 > . As an example of diﬀerential (linear) approximation, we solve #19, 14.4. 2 Week 4 Homework: 14.5 chain rule Problem 14.5.3: Find f ′ (t) when f (x, y ) = sin x cos y, with √ x = x(t) = πt, y = y (t) = t. Solution: We have f ′ (t) = fx x′ + fy y ′ , with fx and fy evaluated at (x, y ) = (x(t), y (t)). Here fx = cos x cos y, fy = sin x(− sin y ), so √ x = x(t) = πt, y = y (t) = t gives √ √ √ fx (πt, t) = cos(πt) cos( t), fy = sin(πt)(− sin t). 1 Then x′ = π, y ′ = 1 t− 2 , so 2 √ √ 1 f ′ (t) = π cos(πt) cos( t) + 1 t− 2 sin(πt)(− sin t). 2 We note that this calculation gives vector structure to what is otherwise a straight calculus 1 calculation, √ d (sin(πt) cos( t)), dt using the product rule and the 1-variable chain-rule. The vector formulation is as a dot product, of a vector < fx , fy >, with < x′ , y ′ >=< x′ (t), y ′ (t) >, where < fx , fy > is evaluated at (x, y ) = (x(t), y (t)). As a preview, the ﬁrst vector is called the gradient of f . We also solved #13 and #23 of section 14.5. ...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

s11wk07su - Math 23 B Dodson Week 3b Homework 14.4 tangent...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online