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May 14 Mathematics 206 Mr. Haines 2010 Multivariable Calculus Late Final Examination (6) I. Give equations for:
A. The set of all points whose distance from (1, 2, 3) is 5.
B. The tangent line to the curve C parametrized by ↅ䙦ᡲ䙧 㐄 䙦ᡲ, ᡲ⡰, ᡲ⡱䙧^ at the point ↅ䙦1䙧.
C. The equation of the tangent plane at the point (0, -1, 2) to the surface whose equation is ᡶ⡱^ ㎗ 12ᡷ ㎗ 3ᡸ⡰^ 㐄 0.
(4) II. If a = i + j and b = i - 2 j c ompute:
A. The length of a.
B. The dot product of a and b.
(6) III. If F : ጷ⡱^ ፲ ጷ⡱^ with rule Ⅲ䙦ᡶ, ᡷ, ᡸ䙧 㐄 䙦ᡶ⡱, ᡷ⡰, ᡸ䙧.
A. Calculate div F
B. Calculate curl F
(6) VI. Suppose ↈ䙦ᡶ, ᡷ, ᡸ䙧 㐄 䙦ᡶ ㎗ ᡷ, ᡶᡷᡸ⡰䙧 and a = (1, 3, 2).
A) Give the Jacobian matrix of f at a.
B) Give the total derivative of f at a.
(10) VII. Use the Divergence Theorem to evaluate∫∫ F • n d σ
∂ S
, where F = x^2 i +xz^2 j + y^2 k and
‴ᡅ is the surface of the unit cube in the first octant. (S = [ 0 , 1 ]×[ 0 , 1 ]×[ 0 , 1 ]).
(5) VIII. If f : ጷ⡱^ ፲ ጷ with rule ᡘ䙦ᡶ, ᡷ, ᡸ䙧 㐄 3ᡶ⡰^ ㎗ 2ᡶᡷ ㎗ 2ᡸ, calculate H f (1, 1, 1), the Hessian of f at (1, 1, 1).
(20) IX. Let M be the surface parametrized by
ↈ䙦ᡱ, ᡲ䙧 㐄 䙦ᡱ, ᡲ, 9 ㎘ ᡱ⡱^ ㎘ ᡲ⡱䙧; 0 㐉 ᡱ 㐉 1; 0 㐉 ᡲ 㐉 2ᡱ
A. Compute ㄅↈ ㄅう 㐀^
ㄅↈ ㄅぇ.
B. Give a unit vector that is perpendicular to M at the point ↈ䙦1, 1䙧 㐄 (1, 1, 7).
C. Set up but do not evaluate an integral which gives the surface area of M.
(10) X. Evaluate the double integral ∫∫ +
R
x^2 y^2 dA where R is the region in the first quadrant
and bounded by the unit circle x^2 + y^2 = 1 by converting to polar coordinates. The conversion equations are ᡶ 㐄 ᡰ cos ‖ ; ᡷ 㐄 ᡰ sin ‖.
(8) XI. For the quadratic form ᡨ䙦ᡶ, ᡷ, ᡸ䙧 㐄 ᡶ⡰^ ㎘ 2ᡷ⡰^ ㎗ 5ᡸ⡰^ ㎘ 2ᡶᡸ ,
A. give a symmetric matrix S that is the matrix of this quadratic form.
B. By taking determinants and using Sylvester’s Theorem, determine if p is positive definite, negative definite, indefinite, or none of these.
(10) XII. Evaluate the line integral (^) ᔖ〄 Ⅲ䙦ᡶ, ᡷ, ᡸ䙧 · ᡖ∆ where Ⅲ䙦ᡶ, ᡷ, ᡸ䙧 㐄 䙦ᡷ, ᡸ, ᡶ䙧 if C is the
straight line segment from (0,0,0) to (1,1,1) parametrized by ᡕ䙦ᡲ䙧 㐄 䙦ᡲ, ᡲ, ᡲ䙧 for 0 㐉 ᡲ 㐉 1.
