This document is a take-home exam for a mathematics course. It contains 3 sections with multiple questions each: 1) limits of multivariable functions, 2) partial differentiation, and 3) tangent planes and differentials. Students are instructed to show their work and solutions independently in a blue book and submit it by a specified deadline.

1. Mathematics 54 6 October 2010
Fifth Long Exam
This is a TAKE HOME Exam. Put your answers in a CLEAN BLUE BOOK writing in front of the RIGHT
PAGES ONLY. WORK INDEPENDENTLY and show complete solutions. DO NOT consult anyoneelse regarding
this exam. This exam is due for submition by 5:30 pm on 7 October 2010 at MB 106.
I. Limits of Functions of More Than One Variable. [5 points each]
sin x2 + y 2 + z 2
1. Find lim .
(x,y,z)→(0,0,0) x2 + y 2 + z 2
x3 y
2. Show that lim does not exist.
(x,y)→(0,0) 2x6 + y2
II. Partial Differentiation. [5 points each]
∂z
1. Let z = sin(xy) cos(y − 2x). Find .
∂x
∂2u
2. Let u = yex + zey + ez . Find .
∂x∂z
1 2 ∂u
3. Let u = ln(3x2 − 2y + 4z 3 ) with x = t 2 , y = t 3 and z = t−2 . Find .
∂t
II. Tangent Planes and Differentials. [5 points each]
1. Find the equation of the plane tangent to the ellipsoid x2 + y 2 + 4z 2 = 12 at the point (2, 2, 1). [Hint: you
∂z ∂z
may have to solve for and explicitly.]
∂x ∂x
3 2
2. Given f (x, y) = x 2 +y , approximate the value of f (2.02, −1.98) using linear approximation and compare it to
the actual value of f (2.02, −1.98).
3. Approximate the volume of platinum needed to coat a 4cm × 3cm × 3cm box with 0.01cm platinum coating
using total differentials.
1