Download Traditional Problems - Calculus - Exam and more Exams Calculus in PDF only on Docsity!
Math 260 Exam 2 Jerry L. Kazdan
March 13, 2012 12:00 – 1:
Directions This exam has two parts. Part A has 6 short answer questions (7 points each, so 42 points) whilePart B has 4 traditional problems (15 points each, so 60 points). Total: 102 points. Neatness counts.
Closed book, no calculators, computers, ipods, cell phomes, etc – but you may use one 3′′^ × 5 ′′^ card with notes on both sides.
Part A: six short answer questions (7 points each, so 42 points).
- Find a 3 × 3 symmetric matrix A with the property that
〈X, AX〉 = −x^21 + 6x 1 x 2 − x 1 x 3 + 2x 2 x 3 + 3x^22
for all X = (x 1 , x 2 , x 3 ) ∈ R^3.
- Under what conditions on the constants a, b, c, and d is the following matrix A positive definite?
A :=
a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 d
- Let B be an anti-symmetric n × n real matrix, so B∗^ = −B. Show that 〈V, BV 〉 = 0 for all V ∈ Rn^.
- Find the arc length of the segment of the helix X(t) := (cos 3t, 1 − 4 t, sin 3t), for 0 ≤ t ≤ π.
- Find some function u(x, y) that satisfies
∂^2 u ∂x∂y = 4 cos(x + 2y) − 2 xy.
- Let v(s) be a smooth function of the real variable s and let u(x, t) := v(x + 3t). Show that u satisfies the homogeneous partial differential equation ut − 3 ux = 0.
[Part B is on the next page]
Part B: four traditional problems (15 points each, so 60 points).
B–1. In an experiment, at time t you measure the value of a quantity R and obtain:
t -1 0 1 2 R -1 1 1 -
Based on other information, you believe the data should fit a curve of the form R = a+bt^2. a) Write the (over-determined) system of linear equations you would ideally like to solve for the unknown coefficients a and b. b) Use the method of least squares to find the normal equations for the coefficients a and b. c) Solve the normal equations to find the coefficients a and b.
B–2. Find and classify all the critical points of f (x, y, z) := x^3 − 3 x + y^2 + z^2.
B–3. For a certain rod of length π , the temperature u(x, t) at the point x at time t satisfies the heat equation ut = uxx. Find all solutions of the special form
u(x, t) = w(x)T (t) for 0 ≤ x ≤ π
that satisfy the boundary conditions u(0, t) = 0 and u(π, t) = 0 for all t ≥ 0.
B–4. Say the equation f (X) := f (x, y, z) = 0 implicitly defines a smooth surface in R^3 (an example is the sphere x^2 + y^2 + z^2 − 4 = 0). Let P ∈ R^3 be a point not on this surface. Assume Q is a point on the surface that is closest to P. Show that the vector from P to Q is orthogonal to the tangent plane to the surface at Q. [Suggestion: Let X(t) be a smooth curve in the surface with X(0) = Q. Then Q is the point on the curve that is closest to P .]
