Dot Product
Lecture 3. 06/20/2012
If a =< a1 , a2 , a3 > and b =< b1 , b2 , b3 >, then their dot product a b is a number given by
a b = a1 b 1 + a2 b 2 + a3 b 3 .
For twodimensional vectors a =< a1 , a2 > and b =< b1 , b2 >, this is just a1 b1 + a2 b2 .
Quiz 3 Solutions
Math 126B. August 7, 2012
Problem 1. [10 points] Approximate 1/1.1 using T1 (x) for f (x) = 1/(1 + x) based at b = 0.
Find the error bound for this approximation.
Solution. f (0) = 1, f (x) = 1/(1 + x)2 , f (0) = 1, so T1 (x) = 1 x, and
1
Math 126, Sections A and B, Winter 2011, Solutions to Midterm II
1. Answer the following.
(a) (4 points) Below is a graph of the surface z = f (x, y).
Decide if the following partial derivatives are positive or negative.
fx (0.2, 0.1) < 0
fy (1, 2) < 0
fx
Math 126, Sections D and F, Spring 2012, Solutions to Midterm II
1. Answer the following about the vector function
r(t) = 5 cos t, 1 + 3 sin t, 4 sin t .
(a) (7 points) Compute the vectors T(t), N(t) and B(t).
r (t) = 5 sin t, 3 cos t, 4 cos t
r (t) =
2
Math 126, Sections C and D , Winter 2014, Midterm II
February 25 , 2014
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There are 4 questions. The exam is out of 40 points.
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Math 126D
Midterm I
Autumn 2010
# 1. (25 points) The points A(1, 2, 3), B(0, 1, 3), and C(2, 1, 1) determine a triangle. (a) Find the interior
angle of the triangle at vertex A. (b) What is the area of the triangle? (c) What is the equation for the plane
Math 126 Selected Taylor Notes HW Answers Below are answers to some of the homework problems, and in some cases hints for getting there. These answers do not constitute full solutions of the problems, only the nal result. You should supply all the interme
Homework #5 Math 126 In the following problems, find the Taylor series for f (x) based at b. Each series can be derived from the basic series given in Examples 4.2. Give an interval I where the Taylor series converges. In problems 18 as was done in Homew
UW, CHEM 152C, W16
CHEM 152 C EXAM 1
VERSION A
Instructor:
Dr. Anne B. McCoy
NO GRAPHING/TEXTENTRY CALCULATORS ALLOWED
Date:
Fri. Jan 29, 2016
Time:
1:302:20 PM
ONLY CALCULATORS MAY BE USED AS CALCULATORS
(you may not use
Midterm 2 Solutions. July 26, 2012
Problem 1. For a function u(x, y ), its Laplacian is dened by
u = uxx + uyy .
Calculate
u for u(x, y ) = ex cos y . Simplify as much as possible.
Solution.
ux = ex cos y, uy = ex sin y,
uxx = ex cos y, uyy = ex cos y,
so
MATH 126B
Midterm 2
July 26, 2012
Name
Student ID #
Section
HONOR STATEMENT
I arm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized ass
Quiz 1
Math 126B. June 28, 2012
Problem 1. [10 points] Suppose A = (1, 0, 0), B = (0, 1, 0), and C = (0, 2, 1). Find the
angle A of the triangle ABC and its area. Round the angle up to the nearest dergee.
Problem 2. [10 points] Find the plane passing thro
Cross Product
Lecture 4. 06/22/2012
Find a nonzero vector v =< x, y, z > orthogonal to a =< 1, 1, 0 > and b =< 1, 0, 2 >. We
need:
0 = v a = x y, 0 = v b = x + 2z x = y = 2z.
Say, let z = 1. Then x = y = 2. General setting: nd v =< x, y, z > orthogonal to
Lines and Planes
Lecture 5. 06/25/2012
Lines. A line l in R3 is dened by any point P0 = (x0 , y0 , z0 ) on it and its direction, represented
by a directional vector v =< a, b, c >. A point P = (x, y, z ) lies on l i P0 P = r r0 =<
x x0 , y y0 , z z0 > is
Vectors in Three Dimensions
Lecture 2. 06/19/2012
Denition of vectors. The term vector is used to indicate a quantity (e.g. velocity or force)
with both magnitude and direction. A quantity which have only magnitude without direction (i.e.
real numbers) is
ThreeDimensional Coordinate System
Lecture 1. 06/18/2012
On the plane, we can represent any point by a pair of two Cartesian coordinates (x, y ), which are
dened by two perpendicular axes: the xaxis and the y axis. In the space, we have three Cartesian
MATH 126B
Final Exam
August 15, 2012
Name
Student ID #
Section
HONOR STATEMENT
I arm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized
Final Exam Solutions. August 15, 2012
Problem 1. Find Taylor series for f (x) = ex
(a) Write it in sigma notation.
(b) Write the rst four nonzero terms.
(c) Find its interval of convergence.
2 /2
based at b = 0.
Solution. (a) Plug in t = x2 /2 into
et =
n
MATH 126B
Midterm 1
July 10, 2012
Name
Student ID #
Section
HONOR STATEMENT
I arm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized ass
Midterm 1 Solutions. July 10, 2012
Problem 1. Are the lines r = r1 + tv1 and r = r2 + sv2 , where
r1 =< 0, 0, 0 >, r2 =< 1, 1, 0 >, v1 =< 2, 1, 3 >, v2 =< 0, 1, 1 >,
parallel, skew. or intersecting?
Solution. They cannot be parallel, since their direction
Quiz 2
Math 126B. July 17, 2012
Problem 1. [10 points] Find and classify critical points of
f (x, y ) = x4 4x y 2 + 2y.
Problem 2. [10 points] Using the linear approximation for f (x, y ) = xy at (x0 , y0 ) = (1, 1),
nd the approximate value of 1.01 1.02.
Quiz 2 Solutions
Math 126B. July 17, 2012
Problem 1. [10 points] Find and classify critical points of
f (x, y ) = x4 4x y 2 + 2y.
Solution. fx = 4x3 4, fy = 2y + 2. So critical points are the solutions of the system
4x3 4 = 0, 2y + 2 = 0 x3 = 1, y = 1 x =
Quiz 3
Math 126B. August 7, 2012
Problem 1. [10 points] Approximate 1/1.1 using T1 (x) for f (x) = 1/(1 + x) based at b = 0.
Find the error bound for this approximation.
Problem 2. [10 points] Approximate 1/1.1 using T3 (x) for f (x) = 1/(1 + x) based at
Quiz 1 Solutions
Math 126B. June 28, 2012
Problem 1. [10 points] Suppose A = (1, 0, 0), B = (0, 1, 0), and C = (0, 2, 1). Find the
angle A of the triangle ABC and its area. Round the angle up to the nearest dergee.
The angle A (let us call it ) is the ang
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Chem 142 Experiment #6: Kinetics II, Integrated Rate Laws
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