Department of Mathematics 21-259 Calculus in 3-D CARNEGIE MELLON UNIVERSITY W. Hrusa Spring 2003

Review Problems for Test 2

1. Consider the function f defined by f (x, y, z) = xy + y^{2}z.

a) Find ∇^{⇀} f.

b) Find the directional derivative of f at the point (2, 3, −1) in the direction of the vector v^{⇀} = ⟨2, 4, −3⟩ .

2. Suppose that f is a smooth function with the following property: At the point (0, 0, 0) , f decreases most rapidly in the direction of ⟨6, −1, 2⟩ and the rate of change of f in this direction is −5. Find ∇^{⇀} f (0, 0, 0) .

3. Find the tangent plane and normal line to the surface z = x^{2} + 4y^{2} at the point P_{0} = (−1, 2, 17) .

4. Let u^{⇀} = \frac{1}{\sqrt{2}} ⟨1, 1⟩ and v^{⇀} = \frac{1}{\sqrt{2}} ⟨1, −1⟩ .

Assume that f_{x} and f_{y} are continuous. If D_{u^{⇀}}f (2, 1) = 2\sqrt{2} and

D_{v^{⇀}}f (2, 1) = −8\sqrt{2}, find ∇^{⇀} f (2, 1) .

5. Find all points on the surface x^{2} + 4y^{2} − z^{2} = 4 at which the tangent plane is parallel to the plane x + 2y + z = 6.

6. Suppose that w = x^{3}y+y^{2}−4z^{3}, x = rs, y = (r^{2} − s^{2})^{3} , and z = 2r+s.

Use the chain rule to compute \frac{∂w}{∂s}. (Express your answer in terms of r and s alone. Algebraic simplication is not necessary.)

7. Suppose that w = f (\frac{y}{y − x}) where f is continuously differentiable.

Show that

x\frac{∂w}{∂x} + y \frac{∂w}{∂y} = 0, x ≠ y.

8. Assume that u and v have continuous second order partial derivatives

and that

\frac{∂u}{∂x} = \frac{∂v}{∂y} , \frac{∂u}{∂y} = \frac{−∂v}{∂x} .

Show that

\frac{∂^{2}u}{∂x^{2}} + \frac{∂^{2}u}{∂y^{2}} = 0 and \frac{∂^{2}v}{∂x^{2}} + \frac{∂^{2}v}{∂y^{2}} = 0.

9. Assume that w = f (u) , where u = x^{2} + y^{2} and f is a smooth function.

Find an expression for \frac{∂^{2}w}{∂x^{2}} + \frac{∂^{2}w}{∂y^{2}} in terms of u, \frac{dw}{du}, and \frac{d^{2}w}{du^{2}}.

10. Find all local maxima, local minima, and saddle points for the function f defined by

a) f (x, y) = x^{2} + xy + y^{2} + x − y + 2,

b) f (x, y) = 3x^{2} − 3xy^{2} + y^{3} + 3y^{2}.

11. Let R be region described by

0 ≤ y ≤ \sqrt{16 − x^{2}}, 0 ≤ x ≤ 4

Find the absolute maximum and the absolute minimum values of

f (x, y) = x^{2} + y^{2} − 4x − 2y + 1

on R.

12. Use Lagrange multipliers to find all possible maxima and minima of f subject to the given constraint.

a) f(x, y, z) = \frac{2}{3} x^{3} + y − 3z; x^{2} + y^{2} + z^{2} = 10

b) f(x, y) = xy; 9x^{2} + y^{2} = 4 13. Write an equivalent double integral with the order of integration re-

versed.

a) ∫_{0}^{1}∫_{y}^{√\overline{y}}f (x, y) dxdy b) ∫_{0}^{2}∫_{1}^{e^{2x}}f (x, y) dydx c) ∫_{0}^{2}∫_{2x}^{4}f (x, y) dydx

14. Evaluate each of the following.

a) ∫_{0}^{1}∫_{0}^{x^{3}}4x^{2}y dydx

b) ∫_{0}^{1}∫_{x}^{1} 3x\sqrt{x^{2} + y^{2}} dydx