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DAWSON COLLEGE
DEPARTMENT OF MATHEMATICS
FINAL EXAMINATION
CALCULUS-III (201-BZF-05)
May 24, 2012 Time: 14:00-17:00 p.m.
Instructor:R. Fournier and T. Kengatharam
Name: ID:
Instructions:
- Translation and regular dictionaries are permitted.
- Scientific non-programmable calculators are permitted.
- Print your name and ID in the provided space.
- This examination booklet must be returned intact.
This examination consists of 20 questions. Please ensure that you have a complete examination before starting.
1
(1) [5 marks]Given that ln 2 =
n=
(−1)n−^1 n , find an integer^ k^ such that
ln 2 −
∑^ k n=
(−1)n−^1 n <^
(2) [5 marks]Find the interval of convergence of the series ∑^ ∞ n=
(x + 1)n+ 2 n^.
(5) [5 marks]Find the equation of the tangent line to the curve r(t) = (cos t, sin t, t) at the point (− 1 , 0 , π).
(6) [5 marks]Find the point(s) on the curve with equation r(t) = (cos t, sin t, t) at which the curvature κ = |r
′×r′′| |r′|^3 is maximal.
(7) [5 marks]Find the arc-length parametrization for the curve r(t) = (cos t, sin t, t), t ≥ 0.
(8) [5 marks]The binormal B(t) is defined as B(t) = T (t) × N (t), where T (t) is the unit tangent vector and N (t) is the unit normal vector of a smooth curve C at any point r(t) ∈ C. Prove that B(t) and B′(t) are perpendicular.
(10) [5 marks]Compute all partial derivatives of order one if
f (x, y, z) =
√^1
x^2 + y^2 + z^2
(11) [5 marks]Find the maximal rate of change and the direction at which it occurs for the function f (x, y, z) = x + xy + xyz at (1, 2 , 3).
(12) [5 marks]Find the maximum of the function f (x, y, z) = xyz under the con- straints g(x, y, z) = x + y + z = 1 and x, y, z > 0. Conclude that for A, B, C > 0, √ (^3) ABC ≤ A + B + C
(13) [5 marks]Find the equation of the tangent plane to the surface with equation x + y^2 + z^3 = 3 at the point (2, 1 , 0).
(15) [5 marks]Prove that f (x, y) = xex^ cos y − yex^ sin y is a solution of the partial differential equation ∂^2 f ∂x^2 +^
∂^2 f ∂y^2 = 0.
(16) [5 marks]Compute the integral ∫ ∫
R
y + xy 1 + y^2 dA where R is the rectangle [0, 2] × [0, 1].
(19) [5 marks]Use polar coordinates to compute the volume of the region lying below the cone with equation z =
x^2 + y^2 and above the disc with equation x^2 + y^2 ≤ 1.
(20) [5 marks]Evaluate ∫ ∫ ∫
E
xe(x^2 +y^2 +z^2 )^2 dV
where E is the upper hemisphere {(x, y, z) | x^2 + y^2 + z^2 ≤ 1; z ≥ 0 }. (You may use the spherical polar coordinates: x = ρ sin ϕ cos θ, y = ρ sin ϕ sin θ, z = ρ cos ϕ, dV = ρ^2 sin ϕdρdϕdθ).
