Assignment 2
Submission Deadline: 18 November 2016 (before 5 p.m.)
Rules:
Collaboration on problem sets is encouraged, but:
Attempt each problem yourself first and foremost. Read each portion of the problem.
If you don't understand the question or what i
Tutorial 8: Applications of Definite Integrals
1. Find the volumes of the solids in:
a. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 1. The
cross-sections perpendicular to the x-axis between these planes are circular disks
wh
Assignment 1
Submission Deadline: 20 October 2016 (before 6 p.m.)
Rules:
Collaboration on problem sets is encouraged, but:
Attempt each problem yourself first and foremost. Read each portion of the problem.
If you don't understand the question or what is
APPLICATIONS OF
DEFINITE INTEGRALS
Calculus I
2
Volumes Using Cross-Sections
A cross-section of a solid is the plane region formed by
intersecting with a plane.
3
Suppose we want to find the volume of a solid .
We begin by extending the definition of a
Assignment 1
Question 1
Using the following graphs, evaluate each expression, or explain why it is undened:
a) (f 0 9X2) (0 (9 9X2)
b) (g f)(2) 6) (f +9)(2)
c=) (ff)(2) f) (f/g)(2)
(09 C 53mm): gm): i
097 Cgo Kg) c2) = W) = l
(C? (83-?) 03 9) l
C? Campe
Chapter 7: Integration
1. Evaluate the following integrals:
a. 2(2 + 4)5
1
b.
(1+)3
27
p. 1 4/3
4
r.
3
1
e. 4 (7 10)
f.
t.
1
3
g. ( + 5)( 5)
h.
3(21)2 +6
3
s. 02 3 (1 + 9 4 )2
( 1)10
(21) cos 3(21)2 +6
1 36
0 (2+1)3
q. 1
c. 37 3 2
d. tan2 sec 2
Tutorial 3
1. (a) Calculate the charge stored on a 3-pF capacitor with 20 V across it.
(b) Find the energy stored in the capacitor.
2. Two capacitors ( 25F and 100F) are connected to a 100-V source. Find the energy stored in
each capacitor if they are con
Tutorial 2
1. For the below circuit, find the branch currents I1, I2 and I3 using mesh analysis.
Answer:
2. Calculate the mesh currents I1 and I2 of the circuit:
Answer:
3. Use mesh analysis to find the current I0 in the circuit:
Answer:
4. Using mesh ana
Runge-Kutta method
The formula for the fourth order Runge-Kutta method (RK4) is given below. Consider the
problem
(
y 0 = f (t, y)
y(t0 ) =
Define h to be the time step size and ti = t0 + ih. Then the following formula
w0 =
k1 = hf (ti , wi )
k1
h
k2 =
4
Linear Multistep Methods
An important class of numerical methods for initial value problems is that of the linear multistep
methods. Assuming the solution has been computed for some number of time steps, the idea is to
use the last k step values to appr
Chapter 8
Runge-Kutta Methods
Main concepts: Generalized collocation method, consistency, order conditions
In this chapter we introduce the most important class of one-step methods that are generically
applicable to ODES (1.2). The formulas describing Run
Appendix A
Runge-Kutta Methods
The Runge-Kutta methods are an important family of iterative methods for the approximation of solutions of ODEs, that were develoved around 1900 by the german
mathematicians C. Runge (18561927) and M.W. Kutta (18671944). We
8.19: Multistep Methods
Multistep methods are methods, which require starting values from several
previous steps.
There are two important families of multistep methods
Adams methods (explicit: AdamsBashforth as predictor, implicit:
AdamsMoulton as correc
Linear Multistep Methods
Lei Liu
October 1, 2008
Model Problem
Linear Multistep Methods
Convergence Analysis
Application to A-D-R equation
Table of Contents
1
2
3
4
Model Problem
Linear Multistep Methods
Definition
The Order Conditions
Classification and
Complex Analysis
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Residues and Poles
Chapter
Resid
Complex Variables Calculus
Chapter Complex Number
Leonhard Euler
De Moivre
Abraham De Moivre was a French
Protestant who moved to England
in search of religious freedom.
He was most famous for his work
on probability and was an
acquaintance of Isaac Newto