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9 Practice Questions for Calculus III - Old Second Test | MATH 320, Exams of Calculus

Material Type: Exam; Class: Multivariate Calc; Subject: Mathematics; University: The University of Tennessee-Martin; Term: Fall 2005;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Math 320 (Fall 2005) Second Test ______________________
Calculus III name (8 points)
This pleasant 50 minute test covers sections 12.7, 13.1-4, 14.1-2 of Calculus by Stewart
5th ed. Relax and do well. Unless otherwise stated, problems are 4 points.
1. Evaluate the integral
π
0
cos t i + (1+t2) k dt
2. Write the following equations in the coordinate system stated.
a) ρ cos φ = ρ sin φ cos θ in rectangular coordinates
b) (x2 + y2)3 = xy in cylindrical coordinates
3. Sketch the curve r(t) = < sin t, sin t, 2 cos t > and indicate the direction in which t
increases.
pf3
pf4

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Math 320 (Fall 2005) Second Test ______________________ Calculus III name (8 points)

This pleasant 50 minute test covers sections 12.7, 13.1-4, 14.1-2 of Calculus by Stewart 5th ed. Relax and do well. Unless otherwise stated, problems are 4 points.

  1. Evaluate the integral ⌡ ⎮

π

cos t i + (1+ t^2 ) k dt

  1. Write the following equations in the coordinate system stated.

a) ρ cos φ = ρ sin φ cos θ in rectangular coordinates

b) ( x^2 + y^2 )^3 = xy in cylindrical coordinates

  1. Sketch the curve r ( t ) = < sin t , sin t , 2 cos t > and indicate the direction in which t increases.
  1. A particle moves with the position function r ( t ) = 2 cos t i + sin t j + sin t k. Find the following.

velocity v ( t )

speed v(t)

acceleration a ( t )

unit tangent vector T ( t )

the derivative T’ ( t )

unit normal vector N (0) (note t = 0)

binormal vector B (0)

  1. Find the domain of the function of f( x,y,z ) =

e x

(^2) + y (^2) − 4

z x^2 + y^2 − 9

  1. Find the following limits or show that they do not exist.

a.

lim ( x,y )→(0,0)

xy x^2 + y^2

b.

lim ( x,y,z )→(1,−1,1)

xyz x + y + z

  1. Find the length of the curve r = cos t i + sin t j + t k from t = 0 to t = π. (8 points)