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ENSC120 Lab Experiment 2
Getting to know the Function Generator and Oscilloscope
Please watch the video tutorial of Oscilloscope and read this document completely and thoroughly before you
even touch any instrument.
Always handle the instruments gentl
Math 254
Solutions - Blue
Midterm
1. [6 marks] Determine whether the statement is True or False.
If not otherwise stated, functions in this question are dened on R3
and they have continuous partial derivatives everywhere.
False The region D = cfw_(x, y) :
Math 254
Solutions - Green
Midterm
1. [6 marks] Determine whether the statement is True or False.
If not otherwise stated, functions in this question are dened on R3
and they have continuous partial derivatives everywhere.
True If F is a vector eld then d
15.3 Functions Given by Power Series
1. Quote. Mathematics knows no races or geographic boundaries; for mathematics, the
cultural world is one country.
(David Hilbert, German mathematician, 1862-1943)
2. Problem. Is the function
f (z) =
n=0
zn
n!
continuo
16.2 Singularities and Zeros. Innity
1. Quote. An innity of passion can be contained in one minute, like a crowd in a small
space.
(Gustave Flaubert, French writer, 1821 - 1880)
2. Problem. What can we say about
lim f (z)
z0
if
(a) f (z) =
1
z
(b) f (z) =
16.3 Residue Integration Method
1. Quote. Love is so short, forgetting is so long.
(Pablo Neruda, Chilean poet, diplomat and politician, 1904-1973)
2. Two Reminders.
(a) If z0 is an isolated singular point of f (z) then f (z) has a Laurent series
an (z z0
15.4 Taylor and Maclaurin Series
1. Quote. I intend to live forever. So far, so good.
(Steven Wright, American comedian, actor and writer, 1955-)
2. Problem. Represent the given analytic function f (z) by a power series.
3. Taylor Series. The Taylor serie
14.3 Cauchys Integral Formula
1. Quote. Of course there is no formula for success except perhaps an unconditional
acceptance of life and what it brings.
(Arthur Rubinstein, Polish-American classical pianist, 1887 - 1982)
2. Goal. Show that
x2
e
0
b2
cos
15.1 Sequence, Series, Convergence Tests
1. Quote. Never forget that the most powerful force on earth is love.
(Nelson Aldrich Rockefeller, American businessman, philanthropist, public servant,
and politician, 1908 - 1979)
2. Problem. What is
i
1+ +
2
i
2
15.2 Power Series
1. Quote. Time is not a line, but a series of now-points.
(Taisen Deshimaru, Japanese Buddhist teacher, 1914 - 1982)
2. Problem. For which z C does the power series
1 + z + z2 + z3 + . . .
converge?
3. Power Series. A power series in pow
14.2 Cauchys Integral Theorem
1. Quote. Change your opinions, keep to your principles; change your leaves, keep intact
your roots.
(Victor Marie Hugo, French poet, novelist, and dramatist, 1802 -1885)
2. Goal. Show that
x2
e
0
b2
cos 2bxdx =
e .
2
3. Rem
14.4 Derivatives of Analytic Functions
1. Quote. If Im free, its because Im always running.
(Jimi Hendrix, American musician, singer and songwriter, 1942 -1970)
2. Goal. Show that
b2
cos 2bxdx =
e .
2
x2
e
0
3. Derivatives of an Analytic Function. If f (
16.7 Surfaces Integrals
1. Quote. The good life is one inspired by love and guided by knowledge.
(Bertrand Arthur William Russell, 3rd Earl Russell, British philosopher, logician, mathematician, historian, and social critic, 1872 - 1970)
2. Problem. Find
16.1 Vector Fields
1. Quote. To a mathematician the term vector eld denotes a slightly fancier construction. A vector
eld X on a manifold M is a smooth section of the tangent bundle T M , that is for each point m M
a choice of a tangent vector X(x) Tm M s
14.1 Line Integral in the Complex Plane
1. Quote. Knowledge is of no value unless you put it into practice.
(Anton Pavlovich Chekhov, Russian physician, dramaturge, and author, 1860-1904)
2. Goal. Show that
x2
e
0
b2
cos 2bxdx =
e .
2
3. The Same But A L
13.7 Logarithm. General Power. Principal Value
1. Quote. The less eort, the faster and more powerful you will be.
(Bruce Lee, Chinese American martial artist, actor, philosopher, and lmmaker, 19401973)
2. Problem. Solve the equation ez = i.
3. Denition. F
Math 254
Solutions
Final Exam
1. [5 marks] Calculate the curl and the divergence of the vector eld
F(x, y, z) = 2 sin(xy)i + eyz j + 3y 2 k
at the point (1, 2, 1).
Solution: From
i
curl F(x, y, z) =
j
k
x
y
yz
z
2
2 sin(xy) e
3y
yz
= (6y ye )i 2x cos(xy)k
Math 254
Midterm 2 - Blue - Solutions
Midterm 2
1. [6 marks] Determine whether the statement is True or False.
False If z = 1 + i then |z| = 2 and Argz = 9/4. [By denition,
< Argz .]
x
False If f (z) = z 1 and z = x + iy then Ref = 2
and Imf =
x + y2
y
y
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ENSC120 Lab Exercise 4
Frequency Response and Diode Circuit
The first part of this exercise involves plotting a frequency response of a circuit. Figure 1 shows a simple R-C
(Resistor Capacitor) circuit. You have to plot its frequency response. Follow
Ser# Date and other info printed here
Your name printed here
Bench Number indicated here
ENSC 120 (2015-3) Final: Gain and Phase Angle Measurement
Examination Record This exam duration is 20 minutes
Write legibly. If I cannot read your writing, I will not
Lab2 Experiment and Record submission
Do not start this experiment before you have gone through the Scope Video Tutorial and the
Function Generator Operation instructions
1. Open the Lab2_Record.pdf using FOXIT.
2. Set the FnGen to default settings [See s
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ENSC120 Lab Exercise 5
Measurement of Phase in sinusoidal signals
This final lab exercise will instruct you how to measure the phase difference between two signals observed using
the oscilloscope. Follow the steps as described below.
R = 10K
Input
Out
Assignment 10
Solutions
Section 15. 3
Find the radius of convergence in two ways: (a) directly by the CauchyHadamard formula in Sec 15.2, and (b) from a series of simpler terms by
using Th 3 or Th 4.
5n
zn
8.
n(n + 1)
n=1
5n
Solution: (a) From an =
by the
Assignment 9
Solutions
Section 15.1
2. Is the sequence zn = (3 + 4i)n /n! bounded? Convergent? Find its
limit points. Show your work in detail.
Solution: We note that |3 + 4i| = 5. Let = arctan(4/5). Then
3 + 4i = 5(cos + i sin ) and
zn =
(5(cos + i sin )
Assignment 8
Section 14.3
12. Integrate counterclockwise. Show the details.
C
z2
z
dz,
+ 4z + 3
C is the circle with centre -1 and radius 2.
Solution: We note that
C
z2
z
dz =
+ 4z + 3
C
z
dz
(z + 1)(z + 3)
and that
| 3 (1)| = 2.
z
is not dened at the poi
Assignment 7
Solutions
Section 13.3
8. Determine and sketch the set of all complex number z such that
|z + i| |z i|.
Solution. Let z = x + iy. Then z + i = x + i(y + 1), z i = x + i(y 1)
and
|z + i| |z i|
|z + i|2 |z i|2
x2 + (y + 1)2 x2 + (y 1)2
y 2 + 2
Assignment 6
Solutions
Section 13.5
4. Find Re z, Im z, and |z| if z = e0.61.8i .
Solution: By denition
z = e0.61.8i = e0.6 (cos(1.8) + i sin(1.8) = e0.6 (cos 1.8 i sin 1.8).
Hence
Re z = e0.6 cos 1.8, Im z = e0.6 sin 1.8, and |z| = e0.6 .
10. Write i and
Assignment 5
Section 13.1
6. If the product of two complex numbers is zero, show that at least one
factor must be zero.
Solution: Let z1 = (a, b) and z2 = (c, d) be two complex numbers such
that
z1 z2 = (a, b) (c, d) = (ac bd, ad + bc) = (0, 0).
Suppose t
Assignment 4
Section 16. 7
12. Evaluate
0 y 1.
Solution: From
S
ydS if S is the surface z = 2 (x3/2 + y 3/2 ), 0 x 1,
3
z
z
= x1/2 and
= y 1/2
x
y
we get
ydS =
2
z
y
y
1+
y
S
1 + (x1/2 ) + (y 1/2 ) dA =
D
=
=
dA
2
y
1 + x + ydA
D
1
ydy
0
2
3
+
2
D
1
=
2
z
Assignment 3
Section 16.4
10. Use Greens Theorem to evaluate the line integral along the given
positively oriented curve.
2
(1 y 3 )dx + (x3 + ey )dy, C is the boundary of the region between the
C
circles x2 + y 2 = 4 and x2 + y 2 = 9.
Solution: Let D be