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Translation - Calculus - Exam, Exams of Calculus

These are the notes of Exam of Calculus which includes Traditional Problems, Symmetric Matrix, Property, Conditions, Constants, Matrix, Positive, Anti Symmetric etc. Key important points are: Translation, Regular Dictionaries, Calculators, Integer, Interval, Convergence, Series, Expansion, Power, Curve

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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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.
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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).