1
Section 4.7: The Natural Logarithmic Function: Integration
Practice HW from Larson Textbook (not to hand in)
p. 285 # 1-25 odd, 35-39 odd
Recall the following integral formula:
x
1
dx =
1
dx = ln | x | +C
x
We illustrate this formula in the following e
1
Section 6.2: Trigonometric Integrals
Practice HW from Larson Textbook (not to hand in)
p. 382 # 5-13 odd, 19-31 odd, 51-55 odd
Integrals Involving Powers of Sine and Cosine
5
Example 1: Integrate sin x cos x dx
Solution:
Larger Powers of Sine and Cosine
1
Section 6.1: Integration by Parts
Practice HW from Larson Textbook (not to hand in)
p. 374 # 1-23 odd, 31-35 odd
Integration by Parts
Integration by parts undoes the product rule of differentiation.
Suppose the have two functions u and v. Differentiatin
1
Section 6.6: Indeterminate Forms and LHospitals Rule
Practice HW from Larson Textbook (not to hand in)
p. 412 # 5-37 odd
In this section, we want to be able to calculate limits that give us indeterminate forms
0
such as
and . In Section 2.5, we learned
1
Section 6.7: Improper Integrals
Practice HW from Stewart Textbook (not to hand in)
p. 422 # 5-8, 5-25 odd, 29, 31
Areas of Infinite Extent
Example 1: Determine the area under the graph of f ( x ) =
1
x2
for x 1
Solution:
The type of integral used to com
1
Section 6.4: Partial Fractions
Practice HW from Larson Textbook (not to hand in)
p. 399 # 1-19 odd
Partial Fractions
Decomposes a rational function into simpler rational functions that are easier to integrate.
Essentially undoes the process of finding a
Assignment 9
u5641446
October 6, 2016
(Question 1)
For this section we are trying to find the group that gets sent to 0. And then the smallest set of elements that generates
that group.
(a)
This function f (x, y) f (0, 0) maps every element to its constan
COMP2610/COMP6261 - Information Theory: Assignment 3
Andrew Markovic u5641446
October 16, 2016
COMP2610/6261 Shared Questions
1. (20 pts) Let X be an ensemble with AX = cfw_x 1 , x 2 , x 3 and probabilities pX = (1/4, 1/3, 5/12).
(a) (2 pt) Compute the c
COMP2610/COMP6261 - Information Theory: Assignment 3
Andrew Markovic u5641446
October 16, 2016
COMP2610/6261 Shared Questions
1. (20 pts) Let X be an ensemble with AX = cfw_x 1 , x 2 , x 3 and probabilities pX = (1/4, 1/3, 5/12).
(a) (2 pt) Compute the c
Assignment 10
u5641446
October 14, 2016
Question 1
From lectures we know if a ring has non-trivial idempotents, R
= eR (1 e)R for e idempotent. So we check if our ring Z8 has any
non-trivial idempotents.
22 = 4
32 = 1
42 = 0
52 = 1
62 = 4
72 = 1
Hence it
Assignment 11
u5641446
October 20, 2016
Question 1
(a)
We know (1) is a maximal ideal. Since our ring is a field, we know every ideal is generated by a monic polynomial. We
want to define a map : R[x] R[x]/(x2 ). We define the map such that constants get
1
Section 5.6: Differential Equations: Growth and Decay
Practice HW from Larson Textbook (not to hand in)
p. 364 # 1-7, 19, 25-34
Differential Equations
Differential Equations are equations that contain an unknown function and one or more
of its derivativ
1
/Section 5.2: Volume: The Disk Method
Practice HW from Larson Textbook (not to hand in)
p. 322 # 1-27 odd
Solids of Revolution
In this section, we want to examine how to find the volume of a solid of revolution,
which is formed by rotating a region in a
1
Section 4.1: Antiderivatives and Indefinite Integration
Practice HW from Larson Textbook (not to hand in)
p. 224 # 1-25 odd, 41-47 odd, 51
Antidifferentiation or Integration
Suppose we are given a derivative of a function f x) 3 x 2 and asked to find f
1
Section 4.5: Integration by Substitution
Practice HW from Larson Textbook (not to hand in)
p. 269 # 1-51 odd, 63, 69-73 odd, 79, 85
Integration by Substitution
2
6
Example 1: Integrate 2 x( x + 5) dx
Solution:
Fact: Integration by substitution undoes th
1
Section 4.6 Numerical Integration
Practice HW from Larson Textbook (not to hand in)
p. 278 # 1 13 odd
Note: Many functions cannot be integrated using the basic integration formulas or with
any technique of integration (substitution, parts, etc.).
2
Exam
1
Section 4.8: Inverse Trigonometric Funtions: Integration
Practice HW from Larson Textbook (not to hand in)
p. 291 # 1-29 odd
Recall that the inverse sine function determines the angle that one must take the sine of to
obtain a given quantity.
Notation:
1
Section 4.4: The Fundamental Theorem of Calculus
Practice HW from Larson Textbook (not to hand in)
p. 257 # 5-35 odd, 43-47 odd, 73-77 odd
Definite Integral
The definite integral is an integral of the form
b
f ( x) dx .
a
This integral is read as the i
1
Section 5.1: Area of a Region Between Two Curves
Practice HW from Larson Textbook (not to hand in)
p. 312 # 1-5 odd, 9, 13-23 odd
Areas Between Curves
Suppose we are given a continuous function y = f (x) where f ( x) 0 over the interval
[a, b].
Recall:
1
Section 5.4: Arc Length and Surfaces of Revolution
Practice HW from Larson Textbook (not to hand in)
p. 340 # 1-13 odd
Arc Length
If we are given two points ( x1 , y1 ) and ( x 2 , y 2 ) on a graph.
( x2 , y 2 )
y
y2
y1
( x1 , y1 )
x1
Distance of Line C
1
Section 5.3: Volume: The Shell Method
Practice HW from Larson Textbook (not to hand in)
p. 330 # 1-9 odd, 13-27 odd
The Shell Method
Uses cylindrical shells to find the volume of a solid of revolution (see Figures 5.27, 5.28)
on p. 325
Horizontal Axis o
1
Section 4.2/4.3: Area, Riemann Sums and the Definite Integral
Practice HW from Larson Textbook (not to hand in)
p. 235 # 1, 3, 19-25, 43 (use Maple), 45 (use Maple) 53, 55
p. 245 # 31-38
Sigma Notation
The sum of n terms a1 , a 2 , , a n is written as
n
1
Section 6.3: Trigonometric Substitution
Practice HW from Larson Textbook (not to hand in)
p. 390 # 5-9 odd, 15-23 odd, 33-37 odd
Trigonometric Substitution
Good for integrating functions with complicated radical expressions.
Useful Identities
1. sin 2 +
Inverse Derivative Problems
1. Let f be the function defined by f(x) = x3 + x. If g(x) = f 1 (x) and g(2) = 1,
what is the value of g (2)?
2. Let f be the function defined by f(x) =
4+ x 5
1
3 x 2 . If g(x) = f (x), and g(3)=2,
what is the value of g (3)?