Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Tests for Convergence of a Series
DEFINITION 1 (Convergence) An infinite series
X
= a1 + a2 + a3 + + an + con
i=1
verges to L if the sequence of partial sums
sn = a1 + a2 + an
converges to a limit L.
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
NAME :
SECTION : (circle one) 12:301;35 1:452:50
Math 8
Practice
Midterm 1
INSTRUCTIONS: This is a closed book exam and no notes are allowed. You are not to provide or
receive help from any outside
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Math 8 Winter 2010 Midterm 2 Review Problems
1
Evaluate
Solutions  1
x cos2 3x dx
Solution: First rewrite cos2 3x using the halfangle formula:
x cos2 3x dx =
x
1 + cos 6x
2
Now use integration by pa
Calculus of Functions of One and Several Variables
MATH 8

Spring 2013
Math 8
Vectors in 2Space and 3Space
Practice Problems
1) Given the vectors a and b
y
b
a
x
sketch in 2Space the following vectors.
a) a + b
b) a b
c) a + 2b
2) Let a =< 5, 12, 1 > and b =< 1, 2, 8
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Harolds Series Convergence Tests
Cheat Sheet
22 March 2013
1
3
2
nth Term Test
Series:
Geometric Series Test
Series:
Condition(s) of Convergence:
None. This test cannot be
used to show convergence
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 5 Integration Review due 04/11/2016 at 10:00am EDT
m8s16
a) 02 3 f (x) dx =
R
b) 06 5 f (x) dx =
R
c) 14 4 f (x) dx =
R
d) 16  3 f (x)  dx =
Solution:
Solution:
R
(a)
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
15.2LimitsandContinuity
1.Definition (Limitof f (x, y)as (x, y) (a, b)
Let f (x, y)beafunctionwhosedomainisD. Suppose (a, b)isa
pointin D suchthatadiskcenteredat (a, b)iscontainedin D.
Wesaythatthelim
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 2 Limits of Taylor Polynomials due 04/01/2016 at 10:00am EDT
third time: D =
fourth time: D =
C. Find an expression, in closed form, for the total vertical
distance the
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 4 Taylor Series as Functions due 04/06/2016 at 10:00am EDT
m8s16
term 3 = 1(x 1)3 .
Thus the series is
1. (1 point) Write the Taylor series for f (x) = ex about
x = 2 a
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 1 The Taylor Polynomial due 03/30/2016 at 10:00am EDT
1. (1 point)
Calculate the Taylor polynomials T2 (x) and T3 (x) centered at
x = for f (x) = cos(x).
Similarly,
T3
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 6 Riemann Sums due 04/11/2016 at 10:00am EDT
m8s16
and the Riemann sum is
1. (1 point) The following sum
s
s
s
2
2
2
2
2
4
2
2n
2
4
+ 4
+.+ 4
n
n
n
n
n
n
h
Area 4
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 7 Solids of Revolution due 04/13/2016 at 10:00am EDT
m8s16
Correct Answers:
1. (1 point) Find the volume of the solid obtained by rotating
the region bounded by
pi*(x7
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 3 Remainder Estimates due 04/04/2016 at 10:00am EDT
m8s16
SOLUTION
By using the Error Bound for Taylor Polynomials, if we approximate sin(1) using the nth degree polyno
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Jonathan B. Kong
Assignment Day 8 Integration by Parts due 04/15/2016 at 10:00am EDT
1. (1 point) Evaluate the definite integral.
Z 8
m8s16
4. (1 point) Find all the values of x such that the given se
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Math 20 Midterm 1 Page 2 of 11
1. [8 points] Determine if the following sequence converges 01 diverges. If
it converges, nd the limit.
{111013 + 71)}00
2
n  1 :1
O) I
Nere \u R 1w} 52/1, ) w 3
was!
Calculus of Functions of One and Several Variables
MATH 8

Winter 2010
Math 8 Practice Exam Problems: This was the rst hour exam from Fall 2000. Our
exam will have a slightly dierent format (50% multiple choice), but the content is roughly
the same.
1. Find the general s