Math 21a: Multivariable calculus
Homework 4: Chain Rule
This homework is due Wednesday, 10/16 rsp Tuesday 10/15.
1 [Compare Stewart 11.5: 2 ] Use the chain rule to nd the derivative
dz/dt for
z = cos(x + 7y) ,
where x(t) = 5t4 and y(t) = 1/t.
Solution:
By
Math 21a: Multivariable calculus
Homework 7: Extrema
1 [Compare Stewart 11.7: 10] Find the local maximum and minimum values and saddle point(s) of the function.
f (x, y) = 5xy +
5 5
+
x y
Solution:
5
The gradient of f is f (x, y) = 5y x2 , 5x y52 . This i
Math 21a: Multivariable calculus
Homework 3: Linearization
[Compare Stewart 11.4: 12,14] a) Find the linearization L(x, y) of
the function f(x, y) = x6y7, at the point (1, 1). b) Find the
linearization L(x, y) of f (x, y) = x + e4y at (3, 0).
Solution:
(a
Math 21a: Multivariable calculus
Homework 8: Lagrange multipliers
1 [Compare Stewart 11.8: 6] Use Lagrange multipliers to nd the
maximum and minimum values of the function subject to the
given constraint(s).
f (x, y) = 3exy ; x3 + y 3 = 16 .
Solution:
By
Math 21a: Multivariable calculus
Homework 11: Polar integration
1 [Compare Stewart 12.3: 54] This is a
review from the last section. Express
the region D bound by the four curves
x = 1, y = 1, y = (x + 1)2, x =
y y 3 as a union of type I or type II
region
Math 21a: Multivariable calculus
Homework 9: Global extrema
1 [Compare Stewart 11.8: 18 ] Find the extreme values of f on the
region described by the inequality.
f (x, y) = 2x2 + 3y 2 4x 5, x2 + y 2 16
Solution:
We need to use Lagrange multipliers for the
Math 21a: Multivariable calculus
Homework 23: Surface area
1 [Compare Stewart 12.6: 12] Find the area of the surface given by
2 3/2
z = (x + y 3/2 ), 0 x 1, 0 y 1 .
3
Solution:
We compute: zx = x1/2 and zy = y 1/2 . The surface area is
1 1
1
0 0
2
3
1
cfw
Math 21a: Multivariable calculus
Homework 13: Triple integrals
1 [Stewart 12.7: 28] Evaluate the iterated integral
2 2y 4y 2
0 0
0
4dx dz dy .
Solution:
E = cfw_(x, y, z) | 0 y 2, 0 z 2 y, 0 x 4 y 2 .,
the solid bounded by the three coordinate planes, the
Math 21a: Multivariable calculus
Homework 6: Directional Derivatives
This homework is due Monday, 10/21 rsp Tuesday 10/22.
1 [Stewart 11.6: 10 ] Find the gradient of
f (x, y, z) = x + yz .
Evaluate the gradient at the point P = (1, 3, 1). Find the rate of
Math 21a: Multivariable calculus
Homework 5: Tangent lines and planes
This homework is due Friday, 10/18 rsp Tuesday 10/22.
1 [Warmup] The equation f (x, y) = x4 y + 5xy 5 = 26 denes a
curve in the xy-plane.
a) Find the slope m of the curve at (2, 1) usin