University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATA37H3 Calculus II for Mathematical Sciences
Examiner: K. Smith
Date: April 25, 2015
Duration: 3 hours
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
SIGNAT
Numerical Integration
Select an integer n, called the number of steps. Divide the
interval of integration, a x b into n equal subintervals, each
of size x =
b a
n.
Decompose
y
y = f (x)
a = x 0 x1 x2 x3
b
a
x1
x0
xj
xj 1
f (x) dx
f (x) dx + +
f (x) dx +
Cylindrical Shells Example
Find the volume of the solid obtained by rotating the region bounded by x = 4 y 2 and x = 8 2y 2
about y = 5.
Solution. The region bounded by x = 4 y 2 and x = 8 2y 2 is sketched below. Note that the two
parabolas meet when 4 y
Trig Functions
Denitions
A
C
C
csc =
A
B
C
C
sec =
B
sin =
C
A
cos =
A
B
B
cot =
A
tan =
B
Radians
For use in calculus, angles are best measured in units called radians. By denition, an arc of length
on a circle of radius one subtends an angle of radians
Substitutions for Integrating Trigonometric Functions
y
dy
sin x
cos x
tan x
cot x
sec x
csc x
cos x dx
sin x dx
sec2 x dx
csc2 x dx
sec x tan x dx
csc x cot x dx
good for
sin x = y, cos2 x = 1 y 2
sin2 x = 1 y 2 , cos x = y
tan x = y, sec2 x = 1 + y 2
Trig Identities Cosine Law and Addition Formulae
The cosine law
If a triangle has sides of length A, B and C and the angle opposite the side of length
C is , then
C 2 = A2 + B 2 2AB cos
Proof:
Applying Pythagorous to the right hand triangle of the right
The RC Circuit
The RC circuit is the electrical circuit consisting of a resistor of resistance R, a capacitor of capacitance C and a voltage source arranged in series. If the charge on the capacitor is Q and the V C R
current flowing in the circuit is I,
Linear Regression
Imagine an experiment in which you measure one quantity, call it y , as a function of a
second quantity, say x. For example, y could be the current that ows through a resistor when
a voltage x is applied to it. Suppose that you measure n
Mar. 21. Convergence tests for series (9.3)
Innite series are a lot like improper integrals of Type I. However, it is often easier
to evaluate integrals, and, it turns out, we can use integrals to help us determine if
series converge or diverge.
Theorem:
$PVOUFSFYBNQMF5BLFBLCLBOEMFUDCFBOZOPO[FSPDPOTUBOU5IFOUIF-)4DO
XIJMFUIF3)4DOOD
/PUJDFUIFTVNPOUIF-)4JTUFMFTDPQJOHBTMO L
L
MO L
MO L
4POPXFYQBOEUIF
TVNBOETJNQMJGZ"MMUIBUSFNBJOTJTMO
MO O
MO O
5IJTGPMMPXTCFDBVTFNJ.JGPSBMMJOCZEFGPGJOGBOETVQ0OFDBOBMMPXBCMZ
NBOJ
The following questions appeared that on some of the previous
midterms on this content. These are only questions for (optional)
extra practice.
1. Using the Riemann definition of the definite integral, evaluate
Z
3
(x3 6x)dx.
0
2. Evaluate
Z 1
x2 sin x dx
Math 121, Spring 2010
Final Exam, April 24
Name:
SID:
Instructor: Pramanik
Section: 201
Problem Points Score
1
10
2
20
3
20
4
10
5
10
6
15
7
15
8
15
9
35
10
10
TOTAL 150
(extra credit)
1
Instructions
The total time is 180 minutes.
The total score is 150
Faculty of Mathematics
University of British Columbia
MATH 121
FINAL EXAM - Winter Term 2009
Time: 12:00-2:30 pm
Date: April 24 , 2009.
Family Name:
First Name:
I.D. Number:
Question
Signature:
Mark
Out of
1
30
2
20
3
10
4
10
5
10
6
10
7
10
Total
100
THER
Math 121, Spring Term 2012
Final Exam
April 11th ,2012
Student number:
LAST name:
First name:
Signature:
Instructions
Do not turn this page over. You will have 150 minutes for the exam (between 8:3011:00)
You may not use books, notes or electronic devic
Interpolating Splines
Splines are used to interpolate functions.
For simplicity we will only consider the most commonly used cubic splines.
Given a function
f :R!R
tabulated for certain points
x0 < x1 <
< xn
called nodes (or knots) we wish to approximate
Apr. 10/11. Applications of Taylor series; Binomial series
(9.7, 9.8)
Example: Express E (x) :=
places.
Example: Evaluate limx0
x t2
e dt
0
as a Maclaurin series, and nd E (1) to 3 decimal
(ex 1x)2
.
x2 ln(1+x2 )
110
The binomial theorem: if n is a positi
Mar. 16. Polar coordinates and polar curves (8.5)
The polar coordinates of a point (x, y ) in the plane are (r, ) as shown:
r=
x2 + y 2 ,
x = r cos( ),
tan( ) =
y
x
y = r sin( ).
Example: Identify the polar curves r = a = const., = = const., r = sec( ).
8
Mar. 2. Work (7.6)
When a constant force moves an object (in the direction of the force), the work done
by that force is
work = force distance
(which has units of energy). If the force is not constant, but varies with position,
the work done is calculated
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 9
You are expected to work on this assignment prior to your tutorial during
the week of 14th. You may ask questions about this assignment
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 4
You are expected to work on this assignment prior to your tutorial during
the week of February 1st. You may ask questions about this as
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 5
You are expected to work on this assignment prior to your tutorial during
the week of February 8th. You may ask questions about this as
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 7
You are expected to work on this assignment prior to your tutorial during
the week of February 29th. You may ask questions about this a
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 3
You are expected to work on this assignment prior to your tutorial during
the week of January 25th. You may ask questions about this as
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 11
You are expected to work on this assignment prior to your tutorial during
the week of March 28th. You may ask questions about this ass