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Calculus III Test 2, October 2006: Limits, Differentiation, Surfaces, and Vector Calculus , Exams of Calculus

Florida International University (FIU)Calculus
Prof. Louis Roder Tcheugoue Tebou

The test questions for mac 2313 calculus iii, held on october 26, 2006. The test covers various topics including limits, differentiability, surface equations, and vector calculus. Students are required to determine the existence of limits, find partial derivatives, find equations for tangent planes and normal lines, calculate velocities and accelerations, and classify critical points.

Typology: Exams

Pre 2010

Uploaded on 03/10/2009

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MAC 2313 (Calculus III)
Test 2, Thursday October 26, 2006
Name: PID:
Remember that no documents or calculators are allowed during the test. Be as precise as possible
in your work; you shall show all your work to deserve the full mark assigned to any question. Do
not cheat, otherwise I will be forced to give you a zero and report your act of cheating to the
University Administration. Goo d luck.
1. [12] Determine whether each of the following limit exits.
a) lim
(x,y)→(0,0)
xy
x2+ 2y2b) lim
(x,y,z)→(0,0,0)
xyz
x2+y2+z2
2. [12] Let f(x, y) =
xy
x2+y2,(x, y)6= (0,0),
0,(x, y) = (0,0).
Find fx(0,0), and fy(0,0). Is fdifferentiable at (0,0)?
pf3
pf4

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MAC 2313 (Calculus III) Test 2, Thursday October 26, 2006

Name: PID:

Remember that no documents or calculators are allowed during the test. Be as precise as possible in your work; you shall show all your work to deserve the full mark assigned to any question. Do not cheat, otherwise I will be forced to give you a zero and report your act of cheating to the University Administration. Good luck.

  1. [12] Determine whether each of the following limit exits.

a) lim (x,y)→(0,0)

xy x^2 + 2y^2

b) lim (x,y,z)→(0, 0 ,0)

xyz x^2 + y^2 + z^2

  1. [12] Let f (x, y) =

xy x^2 + y^2

, (x, y) 6 = (0, 0),

0 , (x, y) = (0, 0). Find fx(0, 0), and fy (0, 0). Is f differentiable at (0, 0)?

  1. [10] Consider the surface xz + 2yz^2 − zy^2 = 1. a) Find an equation for the tangent plane to the surface at the point P (1, 2 , 1). b) Find the parametric equations of the line normal to the surface at P.
  2. [18] a) Find the velocity, the speed, and the acceleration, all of them at time t = π/2, of a particle moving along the curve r(t) = et−→ i + et^ sin(t)−→ j + et^ cos(t)

k. b) What is the curvature of that curve at each time t?

  1. [20] Let f (x, y) = −x^3 + 6x + y^4 − 2 y^2. Find all the critical points of f and classify each of them as a local maximum, a local minimum, or a saddle point.