MATA33 Assignment 8 Winter 2022
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Department of Computer & Mathematical Sciences
MATA33
Assignment 8
Winter 2022
Problems: In Problems 1 - 3, use a Chain Rule to find the indicated partial derivatives.
1. Given is w = 3x2 + xy and x = 2r _ 3s and y = 6r + s. Find ∂w and ∂w
3y ∂r ∂s .
3. Given is w = x2 + ←x + zy2 and x = r2 _ s2 and y = rs and z = r2 + s3 . Find and .
4. Given is z = ←x2 + y2 and x = e2t and y = e −2t. Use a chain rule to find evaluated at
t = 1.
5. Assume w = u(s, t) = f (ss _ t2 , t2 _ s2 ) for where f is a MATA33S function defined for all real s and t. Verify that t us (s, t) + s ut (s, t) = 0.
6. For the following two functions find all point(s) (x, y) such that fx (x, y) = fy (x, y) = 0 (a) f (x, y) = x2 + 2y2 _ x2y
a3 b3
x y
7. Verify that the function z = xey + yex is a solution to the equation zxxx + zyyy = xzxyy + yzxxy
8. Throughout this problem consider n real variables x1 , x2 , ..., xn and let u(x1 , x2 , ..., xn ) = n n
ak xk where ak is a constant for each k = 1, 2, ..., n and ak(2) = 1.
k=1 k=1
n ∂2 z
k=1 ∂xk(2)
9. Let z = x2 + xy + y2 , x = s + t, and y = st. Find and two ways:
(a) By first substituting x and y as functions of s and t and differentiating directly. (b) By the chain rule.
10. Repeat problem 7 for the functions z = , x = set , and y = 1 + se−t
2022-03-29