1. Suppose f (x, y) has values and partial derivatives as in the table at right. Find all the critical points you can
from the given data and classify them into local mins, local maxes, and saddles. (3 points)
Accumulated Data including rst quiz and rst two tests. There are 250 possible points. After the information appears the info for the test given on 10-10. The accumulated point totals are more accurate and more relevant. Accumulated points: Score How many
Math 241 Solutions 7
6.1#7. Note that F(x, y ) = (y + 2, x), x(t) = (sin t, cos t), 0 t /2, so x (t) = (cos t, sin t) and hence
F ds =
F(x(t) x (t)dt =
( cos t + 2)(cos t) + (sin t)(sin t)dt = 2.
6.2#1. Note that F(x, y ) = (x2 y, xy 2 ). Par
Math 241 Solution 3
2.1#44. Since y + 2 here is just shifting the graph to the negative y -axis by 2, we can just think of z = 4x2 + y 2 . Its a paraboloid having an ellipse, 4x2 + y 2 = c, as its section curve at height c. So the original graph is the pa
Accumulated Data including rst quiz and rst three tests. There are 350 possible points. After the information appears the info for the test given on 11-7-08. The accumulated point totals are more accurate and provide a good approximation of your grade to
Accumulated points including both quizzes (including the one given Dec. 1) and all three hour exams. There are 410 possible points. The accumulated point totals provide a good approximation of your grade to this point. These totals do not include your dis
Uses of the gradient John P. DAngelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801 firstname.lastname@example.org denition Let f : R3 R be a function. The partial derivatives of f are dened as usual. For example f (x + h, y, z ) f (x, y,
Practice exam for Midterm 3 in Math 241
Important note: Several of the problems ask you to completely setup but not evaluate a certain
integral. This means that all the limits of integration are specied, and the integrand is in terms of
the nal variables.
1. Let C denote the curve pictured atZ
right, with the orientation shown.
(a) For F(x, y) = x y, 0 , compute F d r directly. (3 points)
F dr =
(b) Check your answer to part (a) using Greens Theorem. (3 points)
2. For each functio
1. For the region R at right, evaluate
2x dA. (4 points)
2x dA =
2. Consider the ellipse C given by x 2 x y + y 2 = 1. Find all the points on C which are closest to the origin. (6
3. For each function
Practice Final Exam for Math 241
1. Consider the points A = (2, 0, 1) and B = (4, 2, 5) in R3 .
(a) Find the point M which is halfway between A and B on the line segment L joining them.
(b) Find the equation for the plane P consisting of all point
1. Suppose f (x, y) has values and partial derivatives as in the table at right.
(a) Find all the critical points you can from the given data and classify them into local mins, local maxes,
and saddles. (3 points)
Math 241 F1H, Spring 2012
Line Integrals: Summary of Notations and Formulas
Curves in Rn :
This is a mostly a recap of material from Chapter 13.
Specifying a curve: The usual way to specify a curve is via a parametrization ~r(t), a vector function that t
Math 241 CD3/CD4 Quiz 1
1. (2 points) Let f (x, y, z ) = (2xz, 0, 3y ). Find the curl and divergence of f . curlf = f =
, , x y z
(2xz, 0, 3y ) = k
(3y ) z
i divf = f =
= (3, 2x, 0).
2. Given a path x(t)
Math 241 Exam 4 Version 1 1. (a) (6pts) Evaluate
yz dx + xz dy + xy dz when r(t) = (t, t2 , t3 ) and t [0, 3]. F ds and use it to
(b) (4pts) Give one of the physical interpretation of the line integral explain the eects of a reparametrization of the p