College of the Holy Cross, Spring Semester, 2013
Math 241, Final Exam Practice Questions
1. Do the points (1, 2, 3), (2, 1, 1), (1, 3, 4) and (1, 4, 2) lie in a common plane? Use any
method, but be sure to explain your reasoning.
2. (a) Let a = 1, 3, 5 an
College of the Holy Cross, Spring Semester, 2013
Math 241, Solutions to Final Exam Practice Questions
1. Do the points (1, 2, 3), (2, 1, 1), (1, 3, 4) and (1, 4, 2) lie in a common plane? Use any
method, but be sure to explain your reasoning.
Let a = 3, 1
Math 241: Multivariable Calculus
Fall 2008
Maple Lab 2: Graphs, Level Curves and Gradient Vector Fields
Professor Levandosky
Instructions: First, nd a partner, log on to a computer and start up Maple. Be sure to
actually log in (do not select workstation
Math 241: Multivariable Calculus
Fall 2008
Maple Lab 1: Plotting Curves, Surfaces and Tangents
Professor Levandosky
Instructions: First, nd a partner, log on to a computer and start up Maple. Be sure to
actually log in (do not select workstation only), so
College of the Holy Cross, Fall Semester, 2005
Math 241
Homework 8: Selected Solutions
11.7.16 The partial derivatives of f are
2 y 2
(2xy 2x3 y )
2 y 2
(x2 2x2 y 2 ).
fx = ex
fy = ex
Since the exponential is always positive the critical points satisfy
2x
Math 241: Multivariable Calculus
College of the Holy Cross, Spring 2013
Homework 3 Answers
1. Consider the quadrilateral with vertices (0, 0), (3, 6), (6, 2) and (2, 1).
(a) Sketch the quadrilateral.
There are actually three dierent quadrilaterals with th
Math 241: Multivariable Calculus
College of the Holy Cross, Spring 2013
Homework 2 Answers
1. Let a = 2, 1 and b = 2, 1.
(a) Compute and sketch a + b together with a and b.
a + b = 4, 0
b
a+b
a
(b) Compute and sketch a b together with a and b.
a + b = 4,
Math 241: Multivariable Calculus
College of the Holy Cross, Spring 2013
Homework 1 Answers
1. Describe and sketch the set of points (x, y, z ) such that z = cos(x).
This is a surface consisting of the graph of the function f (x) = cos(x) in the xz -plane,
Math 241, Midterm Exam 3 Solutions
Prof. Levandosky
1. Use the method of Lagrange multipliers to nd the maximum and minimum values of f (x, y ) =
4x + 5y on the ellipse x2 + xy + y 2 = 21.
Setting f = g gives
4 = (2x + y )
5 = (x + 2y )
Solving both equat
Math 241, Midterm Exam 2 Solutions
Prof. Levandosky
1. Match each function of x and y below with its level curves and its graph.
Function
sin(x) + sin(y )
x2
3x y
Level Curve
(B)
(C)
(F)
Graph
(d)
(f)
(e)
Function
x2 y 2
sin(x + y )
x2 + 2y 2
Level Curve
Math 241, Midterm Exam 1 Solutions
Prof. Levandosky
1. Find the center and radius of the sphere with equation x2 + y 2 + z 2 = 6x 4z .
Complete the square to get (x 3)2 + (y 0)2 + (z + 2)2 = 13. The center is (3, 0, 2)
and the radius is 13.
2. A plane is