Math 440
Preparing for Exam 1
Winter 2006
1
Vocabulary
You should know what the following terms mean and/or how to check whether something fits the given description. Symmetry of a finite shape in R2 Euclidean dot product on Rn Lorentz dot product on R2 L
Math 113, Calculus II
Test 1 Solutions
1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute
0
x2 +
-3
4 3
dx.
Then check your answer using the Evaluation Theorem. 3 3 0 - (-3) = , so xj = a + j x = -3 + j = Solu
Math 113, Calculus II
Winter 2007
Final Exam - Solutions
1. (25 points) Use the limit definition of the definite integral and the sum formulas1 to compute
3 1
3x2 - 4x + 1 dx.
Then check your answer using the Evaluation Theorem. Solution: In this problem,
Math 113, Calculus II
Test 2 - Solutions
1. (25 points) Consider the region bounded by the curves y = x2 and y = 2x - x2 . (a) Sketch the curves and shade in the region bounded by them. 2
y = x2
1 y = 2x - x2
-1 -1
1
2
(b) Find the area of the region abov
Math 113, Calculus II
Final Exam Solutions
1
1. (25 points) Use the limit definition of the definite integral and the sum formulas
2 -2
to compute
4 - x2 dx.
Then check your answer using the Evaluation Theorem. (2) - (-2) 4 Solution: In this integral, we
Math 113, Calculus II 1. (25 points) Consider the region bounded by the curves x 2 = y 3 and y = 1. (a) Sketch both curves and shade in the region. 2
Test 2 Solutions
x2 = y 3
1
y=1
-2
-1 -1
1
2
(b) Find the area of the region above. Solution: Observing t
Test 1 - Solutions
Math 113, Calculus II
February 2, 2007
1. (30 points) Use the limit definition of the definite integral and the sum formulas1 to compute
1 -1
3x2 - 4x + 7 dx.
Then check your answer using the Evaluation Theorem. Solution: In this proble
Calculus II Final Exam Problems - Solutions
1
1. Use the definition of the definite integral to evaluate1
1 -1
(3x - x2 ) dx.
Solution: Recall the limit definition of the definite integral:
b n
f (x) dx = lim
a
n
f (x )x i
i=1
where x = b-a , xi = a + ix
Math 113, Calculus II
Solutions for Test 2: Review
1
Practice Problems - Solutions
1. Determine whether the following series are convergent or divergent. Specify the test you use in each part! (a) n2 + 1 2n2 - 1 Solution: Use the Ratio Test as follows. Co
Math 113, Calculus II
Test 1: Review Solutions
1. Use the limit definition of the definite integral to evaluate
5
(4x - x2 )dx.
2
Solution: By definition, 2 (4x - x2 )dx = limn n f (x )x, where x = b-a = 5-2 = i i=1 n n 3 2 n , f (x) = 4x - x and we may t
Math 113, Calculus II
Test 1: Review
1
Review of the material
Chapter 5 is an introduction to integral calculus. In this chapter we study techniques for evaluating both definite and indefinite integrals of continuous functions.
1.1
SIGMA NOTATION
A conven
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Math 113, Calculus II
Test 2: Review
1
1.1
Integration
Improper Integrals
Improper integrals arise when an infinity or an undefined point appears either in the integrand, the interval, or the integral. Here's what we do:
a f (x)dx
= limb
b a f (x)
dx dx
Math 440, Linear Algebra II
Solutions to Homework 9 Problems
11-5. Let L = R3 . In this problem you will show that L is a Lie algebra via the bracket operation given by the vector cross product: [v, w] = v w for v, w L = R3 by verifying (a)-(c) of Problem
Math 440
Test 2 Solutions
Winter 2006
1
In-class Part
1. (20 points) Let G be the group of matrices G= (You can assume that G is a group.) (a) Define : R G by (t) = 1 t . Prove that is a one-parameter subgroup of G. 0 1 1 a 0 1 : aR .
Proof. To show that
Math 440
Final Exam Solutions
Winter 2006
You can use your notes and any of the materials for this course while working on this exam, but you shouldn't discuss it with anyone but me. This exam is worth 200 points. You can put your completed exam in my mai
Math 440, Linear Algebra II 8-5. Let A= 0 1 . 0 0
Solutions to Homework 7 Problems
(a) Find tA, (tA)2 , (tA)3 , (tA)4 , . . . . 0 t 0 0 Solution: First, tA = , so (tA)2 = = O2 is the zero matrix. We 0 0 0 0 Claim. For all k 2, (tA)k = O2 is the zero matri
Math 440, Linear Algebra II
Solutions to Homework 10 Problems
12-2. The unitary group U (n, C) is the group of linear transformations preserving the Hermitian form on Cn : U (n, C) = cfw_T : Cn Cn | T is linear and T (x) T (y) = x y for all x, y Cn . We s
Math 440
Preparing for Exam 2
Winter 2006
1
Vocabulary
You should be able to define the following things and/or know how to check whether something fits the given description. matrix Lie group, G tangent space at the identity of G, Lie(G) Lie group homomo
Math 440, Linear Algebra II 9-4. Let G = L(2, R), the Lorentz group. Let W =
Solutions to Homework 8 Problems
0 b : bR . b 0
(a) Show that W is a subspace of M(2, R). Proof. Clearly W is a subset of M(2, R), which contains the zero matrix (set b = 0) and
Math 440, Linear Algebra II 6-2. Define a function by
Solutions to Homework 5 Problems
cos(t) - sin(t) 0 (t) = sin(t) cos(t) 0 . 0 0 1 (a) Show that gives a (smooth) curve in the Euclidean group O(3, R) passing through the identity. Proof. First of all,
Math 440, Linear Algebra II
Solutions to Homework 3 Problems
Remark: Recall my practice of labeling problems in the format X-Y, where X refers to the Problem Set in which the problem appears and Y is the number of the problem in Problem Set X. So, for ins
Math 440, Linear Algebra II
Solutions to Homework 4 Problems
4-12. (a) Using (2) and reasoning as in (3) from Problem Set 3, show that L = cfw_M M2 (R) : M T CM = C. Proof. By definition, L(2, R) = cfw_T : R2 R2 | T (v) T (w) = v w for all v, w R2 , where
Math 440, Linear Algebra II
Solutions to Homework 2 Problems
Remark: I'll begin the practice of labeling problems in the format X-Y, where X refers to the Problem Set in which the problem appears and Y is the number of the problem in Problem Set X. For in
Math 440, Linear Algebra II
Solutions to Homework 6 Problems
6-12. Let G = GL(n, R). Show Lie(G) = M(n, R) for n 2. Proof. By Theorem (2.1.1), we know that Lie(G) is a subspace of Mn (R), so it suffices to prove that Lie(G) contains a basis for Mn (R). Le
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