Instructor: Dr. R.A.G. Seely_{(May 2010)} Test 3_{(version A)}

Calculus III (Maths 201–DDB)

Justify all your answers — just having the correct answer is not sufficient. Pace yourself — a rough guide is to spend not more than 2m minutes or so on a question worth m marks.

(6) 1. Suppose F (x, y, z) = x^{2}y − 2x + y − z^{2}.

(a) Find the directional derivative of F at the point P_{0}(1, 2, −11) in the direction from P_{0} to the origin.

(b) For the level surface (contour surface) F (x, y, z) + 119 = 0 find the equation of the tangent plane at P_{0}.

(c) Determine the maximum rate of increase in F at P_{0}; in what direction does the maximum rate occur?

Test 3 2 (4) 2. Suppose z = f(x^{2} − y^{2}, y^{2} − x^{2}), f a differentiable function, show that y \frac{∂z}{∂x} + x\frac{∂z}{∂y} = 0.

(6) 3. Find and classify the critical points of f(x, y) = 4xy − 2x^{4} − y^{2}.

(6) 4. Use Lagrange Multipliers to find the extreme (max and min) values of f(x, y, z) = 3x − y − 3z subject to the constraints z = x + y and x^{2} + 2z^{2} = 18.

Test 3 3 (3×6) 5. Evaluate the following: (change coordinates as appropriate).

(a) ∫_{0}^{1} ∫_{x}^{1} e^{−y^{2}} dy dx
(b) ∫_{0}^{1} ∫_{0}^{√\overline{1−x^{2}}}\frac{1}{1 + x^{2} + y^{2}} dy dx
(c) ∫_{0}^{1} ∫_{0}^{√\overline{1−x^{2}}} ∫_{0}^{√\overline{1−x^{2}−y^{2}}} \sqrt{x^{2} + y^{2} + z^{2}} dz dy dx

Test 3 4 (6) 6. Sketch the solid region S that lies inside both the cylinder x^{2} + y^{2} = 4 and the ellipsoid 2x^{2} +

2y^{2} + z^{2} = 18. Find the volume of S.

(4) 7. Use the transformation { u = x + y , v = x − y } to evaluate the integral ∫∫_{R} e^{x+y} dA, where R is the region bounded by the lines x + y = 0, x + y = 1, x − y = 0, x − y = 1.

(Total: 50)