MATH 231 Group Work Quiz Related 1. Find and sketch the domains of the following functions: (a) f (x, y) = x + y (b) f (x, y) = x + y (c) f (x, y) = x - ln(y - x) 2. Compute the limits or explain why they don't exist. (a) x4 - y 4 (x,y)(0,0) x2 +
MATH 231 Group Work Quiz Related 1. Compute T(t), N(t), B(t) and (t) for the following curves: (a) r(t) = t3 j - 4t k (b) r(t) =
1 2
sin t i +
1 2
cos t j + 1 t k 2
Answers: (a) There was a terribly cruel typo on this problem which makes doing th
Mathematics 231 4th Hour Spring, 2008 Instructor: Tim Emerick Email: tre8a@virginia.edu Office Phone: 924-0899 Office: Kerchof 401 Office Hours: TBA, or by appointment. Structure: The fourth hour will typically have a simple weekly structure: Short
MATH 231 / Spring 2008 Name: Key 1. (4 points) Convert the point (x, y, z) =
1 1 1 2, 2, 2
April 2, 2008
into cylindrical and spherical.
Solution: For cylindrical, we're already part of the way there: we know z. Thus, we only need 2 1 1 2 compute
MATH 231 / Spring 2008 Name: Key 1. (2 points) State Fubini's Theorem. Solution: If f is continuous on the rectangle R = {(x, y) | a x b, c y d}, then
b d d b
March 19, 2008
f (x, y) dA =
R a c
f (x, y) dy dx =
c a
f (x, y) dx dy
2. (8 point
MATH 231 / Spring 2008 Name: Key
March 26, 2008
1. (3 points) Complete the following theorem from the book: if f is continuous on a polar rectangle R given by 0 a r b, , where 0 - 2, then
b
f (x, y) dA =
R a
f (r cos , r sin )r dr d
MATH 231 / Spring 2008 Name: Key
February 6, 2008
1. (4 points) Suppose u and v are differentiable vector functions and f is a real-valued function. Then
d dt [u
(a) (b)
v] = u v + u v = u (f (t)f (t)
d dt [u(f (t)]
2. (6 points) Find a vect
MATH 231 / Spring 2008 Name: Key
February 13, 2008
1. (3 points) For a smooth space curve r(t) with smooth derivative, give the equations for T(t), N(t), and B(t). Solution: T(t) = r (t) |r (t)| N(t) = T (t) |T (t)| B(t) = T(t) N(t)
2. (7 points)
MATH 231 / Spring 2008 Name: Key 1. (3 points) Match the following functions with their graphs. (a) z = sin(xy) (b) z = sin(x - y) (c) z = sin x - sin y
February 20, 2008
(1) Solution: (a)=(1), (b)=(2), (c)=(3).
(2)
(3)
2. (7 points) Find the li
MATH 231 / Spring 2008 Name: Key
January 30, 2008
1. (4 points) Complete the following statements with reference to material from last week's lecture. (a) The vectors a and b are parallel if and only if a b = 0 (b) The vectors a and b are perpendi
MATH 231 / Spring 2008 Name: Key 1. (3 points) What is the equation of a sphere with center (h, k, l) and radius r? Solution: (x - h)2 + (y - k)2 + (z - l)2 = r2
January 23, 2008
- - - - - - - - - - 2. (7 points) If A, B, and C are vertices of
MATH 231 / Spring 2008 Name: Key
March 12, 2008
1. (3 points) Suppose the second partial derivatives of f are continuous on a disk with center (a, b) and suppose that fx (a, b) = fy (a, b) = 0. What does the second derivative test allow us to infer
MATH 231 Group Work Quiz Related 1. Determine whether the points lie on a straight line: (a) (1, 2, 3), (4, 5, 6), and (7, 8, 9). (b) (-1, -5, 1), (1, 2, 5), and (3, 9, 9). Answer: (a) Yes. (b) Yes. 2. Find the unit vector in the direction of the giv
MATH 231 Group Work Quiz Related 1. Find the directional derivative of f at the given point in the direction indicated by the angle or in the direction of the vector v. (a) f (x, y) = x2 + y 2 , ( 1 , 1 ), = 2 2
3 4 .
(b) f (x, y) = ln(x2 + y 2 ),
MATH 231 Group Work Quiz Related 1. Find a b, a b, proja b, and compa b. Can you draw any conclusions about a and b from these results? (a) a = 2, 4, -8 and b = 3, 6, -12 . (b) a = 1, 2, 4 and b = -2, 3, -1 . (c) a = cos t, - sin t, 3 and b = - cos
MATH 231 Group Work Quiz Related 1. Determine whether the points lie on a straight line: (a) (1, 2, 3), (4, 5, 6), and (7, 8, 9). (b) (-1, -5, 1), (1, 2, 5), and (3, 9, 9). 2. Find the unit vector in the direction of the given vector: (a) 2, 0, 0 (b)
MATH 231 Group Work Quiz Related 1. Find and sketch the domains of the following functions: (a) f (x, y) = x + y (b) f (x, y) = x + y (c) f (x, y) = x - ln(y - x) Answers:
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
(a)
2
1
-5
-4
-3
MATH 231 Group Work Quiz Related 1. Find the absolute maximum and minimum values of he function f on the set D. (a) f (x, y) = 3 + xy - x - y 2 , where D is the triangle bounded by the points (0, 0), (2, 0), (1, 1). (b) f (x, y) = 4x + 6y - x2 - y 2
MATH 231 Group Work Quiz Related 1. Classify and sketch the following surfaces: (a) z = 2x2 + y 2 (b) z 2 = x2 + 1 y 2 3 (c) 1 + 5z 2 = 7x2 - y 2 (d) 4 + z 2 = x2 + y 2 2. What is the domain of the vector function r(t) =
-1 2 - t, ln t, tt-1 ?
2
1
MATH 231 Group Work Quiz Related 1. Find the directional derivative of f at the given point in the direction indicated by the angle or in the direction of the vector v. (a) f (x, y) = x2 + y 2 , ( 1 , 1 ), = 2 2
3 4 .
(b) f (x, y) = ln(x2 + y 2 ),
MATH 231 Group Work Quiz Related 1. Evaluate the double integral: (a) (2x + y) dA where D is the region bounded by y =
D
x and y = x2
(b)
1 0 y
2-y 2
(x + y) dx dy (c) 2x2 y dA where D is the triangular region with vertices (0, 0), (1, 2), an
MATH 231 Group Work Quiz Related 1. Compute T(t), N(t), B(t) and (t) for the following curves: (a) r(t) = t3 j - 4t k (b) r(t) =
1 2
sin t i +
1 2
cos t j + 1 t k 2
2. A gun is fired with angle of elevation 30 . What is the muzzle speed if the ma
MATH 231 Group Work Quiz Related 1. Find the absolute maximum and minimum values of he function f on the set D. (a) f (x, y) = 3 + xy - x - y 2 , where D is the triangle bounded by the points (0, 0), (2, 0), (1, 1). (b) f (x, y) = 4x + 6y - x2 - y 2
MATH 231 Group Work Quiz Related 1. Find a b, a b, proja b, and compa b. Can you draw any conclusions about a and b from these results? (a) a = 2, 4, -8 and b = 3, 6, -12 . (b) a = 1, 2, 4 and b = -2, 3, -1 . (c) a = cos t, - sin t, 3 and b = - cos
MATH 231 Group Work Quiz Related 1. Classify and sketch the following surfaces: (a) z = 2x2 + y 2 (b) z 2 = x2 + 1 y 2 3 (c) 1 + 5z 2 = 7x2 - y 2 (d) 4 + z 2 = x2 + y 2 Answer: (a) Elliptic paraboloid.
(b) Cone.
(c) Hyperboloid of two sheets.
(d)
MATH 231 Group Work Quiz Related 1. Evaluate the double integral: (a) (2x + y) dA where D is the region bounded by y =
D
x and y = x2
(b)
1 0 y
2-y 2
(x + y) dx dy (c) 2x2 y dA where D is the triangular region with vertices (0, 0), (1, 2), an
1. Hello/Xin Cho
A. Dialogs
Sydney: Xin cho. Ti tn l Sydney. Bn tn g?
Matt: Cho Sydney. Ti tn l Matt. Bn khe
khng?
Matt: Hi Sydney, my name is Matt. How are you?
Sydney: Ti khe, cm n, cn bn?
Sydney: I am good, thank you, and you?
Matt: Ti khe. Rt vui c gp