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) The definite integral 02 f (x) dx is the signed area o
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 parts to evaluate
dx =
1
2
x dx +
1
2
x cos 6x dx.
x cos
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 >. Find the following
a) a + b
c) 3a 2b
b) a + b
d) 
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 source during the exam except that you may ask the inst
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.
Wesaythatthelimit f (x, y)as (x, y)approaches (a, b)is L and
write
lim
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 ball has traveled when it hits the floor for the nth
t
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 as
cn (x + 2)n .
n=0
1
= 1 1(x 1) + 1(x 1)2 1(x 1)3 + .
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 (x) = D + E(x ) + F(x )2 + G(x )3
where
D = f () = 1 =
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 1
h.
8
Then the corresponding definite integral gives
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+1)2*Dx
2*pi*[1+0*2/(1+7)+1/(1+2*7)]
x = 3y2 , y = 1,
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 polynomial, the error is at
1
. For the answer to be correct
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 series
would converge.
ln x9 dx
2
n!(x 3)n
n=1
Answer:
C
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!)
M11) Us, 7 {UL )1
A i i \ /'
Hwy(.0
E E% s 5 t
,._}
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 solution to the dierential equation (1 + x2 )y + 2xy = 3