MATH 2220 HW6.
Due Wednesday 15 October
(1) Section 3.5, p. 253-255.
(a) # 8.
(b) # 12.
(2) Review Exercises, p. 255-259.
(a) # 5.
(b) # 15.
(c) # 22.
(3) Find the maximum and minimum values of the function f (x, y ) = ex + ey on the
line segment in R2 jo
MATH 2220 HW11.
Due Friday 5 December
(1) Let S be the surface obtained by rotating the graph of the function
y = |x | + 1
about the xaxis between x = 2 and x = 2.
(a) Let F(x, y, z ) = (0, (x2 2)z, y ). Calculate
curl(F) dS
S
with respect to either orien
MATH 2220 PRELIM 1
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems you u
MATH 2220 PRELIM 1: SOLUTIONS
(1) State whether the following are true or false and justify your answer:
(a) The vectors (5, 7, 1) and (1, 0, 5) are orthogonal.
True, since
(5, 7, 1) (1, 0, 5) = 0.
(b) The angle between the vectors (0, 0, 1) and (5, 7, 1)
MATH 2220 PRELIM (PRACTICE)
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. The exam is printed on both
sides of the paper.
Good luck!
(1) Calculate:
(a)
1
x
sin
MATH 2220 PRELIM 2
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems you u
QUIZ
1. MENTAL
Calculate the following (it is possible to do it without using pen and paper).
(1) The outward ux of the vector eld F(x, y, z ) = (x, 0, 0) through the surface of the
cube [0, 2] [0, 2] [0, 2] with the cube [0, 1] [0, 1] [0, 1] removed.
(2)
MATH 2220 PRELIM (PRACTICE)
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. The exam is printed on both
sides of the paper.
Good luck!
(1) State whether the foll
Math 2220 Exam 2
Thursday, March 29, 2012
Name:
Show all work and explain all answers except as noted.
1. a. If =
x
x2 +y 2 dx
+
y
x2 +y 2 dy ,
calculate d .
b. For each k = 0, 1, 2, 3, describe how k -forms on R3 relate to either vector elds or
scalar fu
Math 2220 Exam 1 review (sample test)
1. a. Find the equation of the tangent plane to the surface dened by
ln(x2 + z 4 1) x2 (y 2) z = 0
at the point (1, 3, 1).
b. Find all values of (a, b) for which the tangent plane to z = x2 y xy 2 x + y + xy at
(x, y
Math 2220 - Prelim #1 Solutions
1. For a xed constant a, let g (t) = (a2 t, at2 , t3 ).
(a) For which values of a is |g (a)|2 = 14?
We have
g (t) = (a2 , 2at, 3t2 ),
so
g (a) = (a2 , 2a2 , 3a2 )
and |g (a)|2 = (a2 )2 + (2a2 )2 + (3a2 )2 = 14a4 . Thus a =
Math 2220 HW 6 Solutions
March 12th, 2014
1. Using double integrals nd the volume of the region enclosed by the coordinate planes and the plane
3x ` 2y ` z 6.
Solution: Our function will be z 6 3x 2y . The region we integrate over in the xy -plane is boun
Math 2220 HW 5 Solutions
March 5th, 2014
1. Let f pu, v q peuv , evu q. For what values of u, v, x and y can f be inverted? Here xpu, v q and y pu, v q
are the rst and second coordinate of f pu, v q. Compute Dpf 1 q where possible.
Solution: Using the inv
MATH 2220 HW10.
Due Wednesday 26 November (or earlier)
(1) Let ai , bi , ci be real numbers. Dene a vector eld F : R3 R3 by
F(x, y, z ) = (a1 x + a2 y + a3 z, b1 x + b2 y + b3 z, c1 x + c2 y + c3 z ).
(a) Calculate curl(F).
Recall that curl(F) is
i
j
k
x
MATH 2220 HW9.
Due Wednesday 19 November
(1) Find
C
F dr where C is the semicircle x2 + y 2 = 1, y 0, oriented anticlockwise,
and F(x, y ) = (y, x).
C may be parametrized as c(t) = (cos t, sin t) with 0 t . So the desired
integral is
( sin t, cos t) ( sin
MATH 2220 HW8.
Due Wednesday 12 November
(1) Section 6.2 p. 392.
(a) # 29.
For part (a), one way to do this is to change coordinates. The easiest choice is
to take spherical coordinates but modify them slightly. Thus, we take
x = a cos() sin()
y = b sin()
MATH 2220 HW5.
Due Wednesday 8 October
(1) Section 3.3 p. 202.
(a) # 5.
(b) # 6.
(2) p. 222 - 225.
(a) # 6.
(b) # 33.
(c) # 41(b).
(3) Section 3.4 p. 243 - 246.
(a) # 5
(b) # 12.
(c) # 30.
(4) Let f : R2 R be a C function with a critical point at (0, 0).
MATH 2220 HW4.
Due Wednesday 24 September
(1) Use the derivative to estimate the value of cos(0.02 cos(0.03).
(2) Section 2.5 p.159-163
(a) # 11.
(b) # 15.
(c) # 24.
(3) Section 2.6 p.171-173
(a) # 4(b).
(b) # 14(a).
(c) # 15.
(4) The surface of a mountai
MATH 2220 HW3.
Due Wednesday 17 September
(1) Find
lim
(x,y )(0,0) x2
x
+ y2
or show that it does not exist.
(2) Section 2.3, p. 139-141
(a) # 1(d).
(b) # 4(b).
(c) # 5.
(d) # 7(c).
(e) # 9.
2
2
2
(3) The temperature at a point (x, y, z ) R3 is given by T
MATH 2220 SYLLABUS AND COURSE INFORMATION 2008
Note: some of the information in this document has not yet been nalized.
General information
Instructor: Dr. Richard Vale, Malott Hall 583.
[email protected]
Lecture 1: MWF 0905-0955 in BKL (Baker Labora
MATH 2220 HW1 AND HW2.
Homework 1. Due Wednesday 3 September.
(1) Section 1.3, p. 6165
(a) # 8.
(b) # 24.
(c) # 36.
(2) Find a unit vector parallel to the line of intersection of the planes x 2y + 5z = 2
and 3x y + 5z = 3.
(3) Given the points P = (1, 2,
MATH 2220 FINAL EXAM
You have 2 hours 30 minutes to complete this exam. The exam starts at 7:00pm. Each
question is worth 20 marks. There are 8 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems yo
MATH 2220 FINAL EXAM: SOLUTIONS
You have 2 hours 30 minutes to complete this exam. The exam starts at 7:00pm. Each
question is worth 20 marks. There are 8 questions in total. You are free to use results from
the lectures, but you should clearly state any
MATH 2220 HW1 SOLUTIONS.
Homework 1. Due Wednesday 3 September.
(1) Section 1.3, p. 6165
(a) # 8.
The volume is given by absolute value of the determinant
10
0
3 1
0 3 1 =
2 1
4 2 1
= 1.
So the volume is | 1| = 1.
(b) # 24.
The direction vector of the lin
MATH 2220 HW2 SOLUTIONS.
Homework 2. Due Wednesday 10 September.
(1) Section 2.1 p. 105107
(a) # 2a.
If c < 1, then the level curve is empty. If c = 1 then the level curve is a single
point (0, 0). If c > 1 then the level curve is a circle with midpoint (
MATH 2220 HW3.
Due Wednesday 17 September
(1) Find
lim
(x,y )(0,0) x2
x
+ y2
or show that it does not exist.
The limit does not exist. For example, if we approach (0, 0) along the path x = 0,
the limit is 0. If we approach along the path y = x, the limit
MATH 2220 HW4.
Due Wednesday 24 September
(1) Use the derivative to estimate the value of cos(0.02 cos(0.03).
Let f (x, y ) = cos(x cos(y ). Let x0 = (0, 0). Let x = (0.02, 0.03). We estimate
f (x) by using the linear approximation:
f (x0 ) +
f (x0 )(x x0
MATH 2220 HW6.
Due Wednesday 15 October
(1) Section 3.5, p. 253-255.
(a) # 8.
The inverse function theorem tells us that we need to check that the matrix
1 + yz xz
xy
y
1+x
0
2
0
1 + 6z
is invertible at (0, 0, 0). Plugging in these values, the matrix beco
Math 2220 HW 3 Solutions
February 12th, 2014
1. Let
5
f px, y q
x y
xy x2 y2
2
if px, y q $ p0, 0q
2
if px, y q p0, 0q.
0
Compute the mixed partials at p0, 0q. Does this contradict the theorem 1 in section 3.1? Justify your
answer.
Solution: Using the qu
Math 2220 HW 4 Solutions
February 26th, 2014
1. Find the absolute maximum of f px, y, z q xy ` yz on the surface x2 ` y 2 ` z 2 1.
Solution: This is a Lagrangian multipliers problem with g px, y, z q x2 ` y 2 ` z 2 1. Thus we need
f g , or
y
2x
x ` z 2y .