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
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
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 ske
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
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
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:
Applyi
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 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
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 hel
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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
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
O
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 8:301
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
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
.
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
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
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 we
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 4
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the we
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 5
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the we
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 7
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the we
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MATA37
Winter 2016
Assignment # 3
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the we
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 w