1. The two curves r1 (t) = 2, t, t2 4 and r2 (s) = s, 3, 9 s2 both lie on a surface S
and intersect at some point P .
(a) Find their point of intersection.
(b) Find the angle between the tangent vectors of the curves at the point of intersection.
(c) Find
Math 11
1 of 3
5
y ex dA =
1. (25) (Show all work) The double integral
D
2x2
1
y exp(x5 )dA can be exD
5
y ex dydx.
pressed as the iterated integral
0
1
(a) Evaluate this iterated integral.
1
1
y2
2
2x2
1
2 5
e dx =
2x e dx = ex
5
1
x5
0
1
4 x5
1
2
= (e e
Math 11 Exam, Fall 2005, Solutions
Problem 1
(a) The direction in which the temperature will rise most rapidly is given
by the gradient, so the direction in which the temperature will fall
most rapidly is given by the negative of the gradient. The tempera
MATH 11: MIDTERM EXAM #1
SOLUTIONS
Problem 1(a). False: if you take u = v = 0, then u v = 0 because the vectors are
parallel, but |u v| = 0 = |u|v|.
Problem 1(b). The angle is acute. Let a = 3, 1, 2 and b = 2, 4, 1 . Then a b =
|a|b| cos where is the angl
NOTE: In no sense should this collection of problems be construed as representative of the
actual exam. These are simply some problems left over from our preparation of the exam or from
previous exams which should serve to indicate the general level of ex
FERPA WAIVER: WRITTEN HOMEWORK
By my signature I relinquish my FERPA rights in the following context: My written
homework sets for Math 11, Fall 2014, may be returned en masse with others in the class
via the homework boxes in the hallway. In addition, ex
Math 11. Multivariable Calculus.
Written Homework 6.
Due on Wednesday, 10/29/14.
You can turn in this homework by leaving it in the boxes labeled Math 11 in the hallway
outside of 008 Kemeny anytime before 3:00 pm on Wednesday.
1. (a) Show that if f is a
Math 11. Multivariable Calculus.
Practice Homework 9.
No due date.
This homework set is not to be turned in. It is for you to practice and prepare for the
exam. Solutions will be posted on the course website.
1. Find the center of mass of the hemisphere x
Math 11. Multivariable Calculus.
Written Homework 5.
Due on Wednesday, 10/22/14.
You can turn in this homework by leaving it in the boxes labeled Math 11 in the hallway
outside of 008 Kemeny anytime before 3:00 pm on Wednesday.
1. Find the gradient vector
Math 11
1 of 6
5
y ex dA =
1. (25) (Show all work) The double integral
D
2x2
1
y exp(x5 )dA can be exD
5
y ex dydx.
pressed as the iterated integral
1
0
(a) Evaluate this iterated integral.
5
y ex dA.
(b) Draw (and shade) the region D corresponding to
D
5
Math 11
Fall 2012
Practice Exam I
This practice exam is intended to give you an idea of the kinds of problems we consider
putting on an exam, and of the possible length of a midterm exam. Any topic that appeared
on a homework problem may appear on the exa
Here are some solutions. As always no guarantee there are no typos, but I think things
are correct.
1. Consider two vector elds F = y, x and G = cos x + y, x 1 dened in the plane.
(a) Determine whether F or G is conservative. If conservative, produce a po
Math 11
1 of 2
NOTE: In no sense should this collection of problems be construed as representative of the
actual exam. These are simply some problems left over from our preparation of the exam or from
previous exams which should serve to indicate the gene
1. The two curves r1 (t) = 2, t, t2 4 and r2 (s) = s, 3, 9 s2 both lie on a surface S
and intersect at some point P .
(a) Find their point of intersection.
(b) Find the angle between the tangent vectors of the curves at the point of intersection.
(c) Find
Here are some problems to keep you awake at night. As usual these are not necessarily
representative of the problems on the nal, but should give you a decent review of the new
material. You are on your own for the old material.
1. Consider two vector elds
Math 11 Midterm Exam #2
October 30, 2014
Instructor (circle one): Hanlon/Zhao, Voight, Sadykov, Elizalde
PRINT NAME:
Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted. You
must justify all of your answers to recei
Mathematics 11
Practice Exam 2
1. Consider the function
f (x, y) = 2xy + x2 .
(a) Find all critical points of f .
(b) For each critical point, determine whether it is a local maximum, local minimum,
or saddle point.
(c) Consider the rectangular region
D =
Math 11
1 of 8
1. (20) (Show all work) Once again the starship Puddlejumper nds itself in trouble. With
engines dead, they have drifted near the sun and nd themselves watching their hull tem2
2
2
perature which changes according to the function T (x, y, z
Math 11. Multivariable Calculus.
Written Homework 7.
Due on Wednesday, 11/5/14.
You can turn in this homework by leaving it in the boxes labeled Math 11 in the hallway
outside of 008 Kemeny anytime before 3:00 pm on Wednesday.
1. Use polar coordinates to
Math 11. Multivariable Calculus.
Written Homework 2.
Due on Wednesday, 10/1/14.
You can turn in this homework by leaving it in the boxes labeled Math 11 in the hallway
outside of 008 Kemeny anytime before 3:00 pm on Wednesday.
1. Show that the curve given
Math 11. Multivariable Calculus.
Written Homework 8.
Due on Wednesday, 11/12/14.
You can turn in this homework by leaving it in the boxes labeled Math 11 in the hallway
outside of 008 Kemeny anytime before 3:00 pm on Wednesday.
1. Let D be a region bounde
Sample Solutions to Practice Problems for
Exam I
Math 11 Fall 2007
October 17, 2008
In these solutions I have shown enough work to get full credit on an exam.
Parenthetical comments are extra, for your benet.
1. TRUE or FALSE: There is a function f : R2 R
M4 at 21 Fax/22015:) SQLUTYan/g/gatv
Important hint: In several, but not all, of the problems below, you can simplify the
work by applying one of the theorems from Chapter 17. Think before you calculate!
If an integral looks impossible, see if you can us
MM 11 M £0 to Maa :3. game.
_ (I _ 59%
1. [8 points] - * N61;- 2 w M fir, hm?
(6;) Find the locations of all local maxima, minima, and saddle points of the function zz, y) =
(3:2 :62)? in the plane. (Be sure to state the type of each point found.)
f ,
Your name: 5 0 [who n S
Instructor (please circle): Barnett Van Erp
Math 11 Fall 2010: written part of HW7 (due Wed Nov 10)
Please show your work. No credit is given for solutions without justication.
(1) [8 points]
(a) Find fff f dV where f(a;, y, z)
We gave-"1006's M-
Your name:
Instructor (please circle): Barnett Van Erp
Math 11 Fall 2010: written part of HW3 (due Wed Oct 13)
Please show your work. No credit is given for solutions without justication.
(1) [10 points] For each of the limits in i
a ---~-W»> é ammo; '--~
Instructor (please circle): Barnett Van Erp
Your name:
Math 11 Fall 2010: written part of HW6 (clue Wed Nov 3)
Please show your work. No credit is given for solutions without justication.
(1) [8 points] Evaluate the following
Yen: name: 32m?
Instructor (please circle): um Van EIp
M_._________,/
Math 11 Fall 2010: written part of HW9 (due Mon Nov 29)
Please Show you?" work. No credit is green for soutéom without jesmlcetéon.
".7 1- "f?
\4 u-
(l) points] --
net F = (2123