Math 1512, First Exam Practice Problem Set
1. Denitions and statements of results:
(a) Dene what it means for a set S of vectors in a linear space V to span V .
(b) State the Cauchy-Schwartz inequality for vectors x, y in a Euclidean space V ,
including t
Math 1512: Honors Calculus II
Fall 2010
Homework for the lecture on Nov 23
1. (a) Assume that ak 0, show that
(ak ak+1 ) = a1 .
k=1
(b) Sum the series
k=1
2k + 1
.
2k 2 (k + 1)2
2. Determine whether each of the following series converges or diverges. Just
Math 1512: Honors Calculus II
Fall 2010
Homework for the lecture on Oct 28
1. Find the least squares regression line for the data points
(0, 10), (2, 6), (3, 7), (4, 6), (5, 3), (8, 1).
2. Let
21
1
4 2 , b = 0 .
A=
11
0
(a) Find the orthogonal projection
Math 1512: Honors Calculus II
Fall 2010
Homework for the lecture on Nov 30
1. Determine whether each of the following series converges or diverges. If the series converges, determine whether it converges absolutely. Justify your answers.
(a)
k=1
k!
1 3 5
Math 1512: Honors Calculus II
Fall 2010
Homework for the lecture on Dec 2
1. (a) Find the Taylor polynomial p2 (x) for the function f (x) = sin(2x) centered around
x = 0.
(b) Use the Lagrange form of the remainder r2 (x) to prove that | sin(1) 1| < 0.2.
2
Math 1512, Final Exam Review (Part 2)
Tue, Dec 14th, 2:50pm-5:40pm, Skiles 271. Oce hours: Mon 3-5pm.
Topics/keywords:
Eigenvalues and eigenvectors: computation in the nite-dimensional case, characteristic polynomial.
Cayley-Hamilton theorem, Jordan can
Math 1512, Final Exam Review (Part 1)
Tue, Dec 14th, 2:50pm-5:40pm, Skiles 271
Topics/keywords:
Vector algebra: the dot and cross products, norm, angle between two vectors, scalar
triple product.
Linear span, linear independence, basis, dimension.
Appl
Math 1512, Second Exam Practice Problem Set
1. Denitions and statements of results:
(a) Describe Gauss-Jordan process of nding a row echelon form of a matrix.
(b) What are elementary row operations? How can one describe these operations
via multiplication