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Second order Differential equations
An equation is termed second order when it contains the second derivative
The solution of a second-order differential equation depends on the type of solution which satisfies its
auxiliary equations. There are three typ
MATH 221
FIRST SEMESTER
CALCULUS
fall 2009
Typeset:June 8, 2010
1
MATH 221 1st SEMESTER CALCULUS
LECTURE NOTES VERSION 2.0 (fall 2009)
This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting
A
from a
NYU-Polytechnic School of Engineering
MA 1124/1424
Review Problems for Final Exam
(1) Given the following gure:
2
f (x)dx =
(a)
3
(b) The average value of f (x) over the interval (0, 3) is
(c) The area of the shaded region is
.
.
(2) Use geometry or basic
NYU - Polytechnic School of Engineering
MA 1124/1424
Review Problems for Exam 3
(1) Given the region bounded by the curves y = x and y = x3 . Find the area of the
region, and also the volume of the solid generated by revolving this region around
the line
NYU - Polytechnic School of Engineering
125 Pre-Calculus Problems
This is a set of practice PreCalc questions that reect material you are expected (1) to
have mastered in high school and (2) to have at your ngertips while taking Math at the
NYU SoE. If yo
NYU-Polytechnic School of Engineering
MA 1124/1424
Review Problems for Exam 1
x
(1) Given the graph of the function g as below. Let G(x) =
g(t) dt. Fill in each of
0
the following blanks.
Then G(5) =
, and the average value of g on [0, 5] is
.
(2) Let f a
NYU-Polytechnic School of Engineering
Review Problems for Exam 2 (Calculus 2)
(1) Evaluate each of the following denite integrals.
1
y(y 2 + 4)6 dy
(a)
0
3
(b)
x(x2 + x + 1) dx
0
3
[|3x 7| + 5] dx
(c)
0
(2) Evaluate each of the following denite integrals.
Harolds Calculus Notes
Cheat Sheet
1 October 2015
AP Calculus
Limits
Definition of Limit
Let f be a function defined on an open interval
containing c and let L be a real number. The
statement:
lim () =
means that for each > 0 there exists a > 0
such that
CALCULUS
Introduction to Limits
Section 10.1
denition of the limit
limits from a graph
limits from an algebraic expression
indeterminate forms
c csun 2009
1
Denition of the limit
Let f (x) be a function, and let c and L be real numbers.
lim f (x) = L,
www.mathportal.org
Limits and Derivatives Formulas
1. Limits
Power rule
Properties
if lim f ( x) = l and lim g ( x) = m , then
xa
xa
lim [ f ( x ) g ( x)] = l m
xa
lim [ f ( x) g ( x) ] = l m
xa
lim c f ( x) = c l
x a
x a
( )
Chain rule
d
f ( g ( x ) ) =
CHAPTER 10
Limit Formulas
10.1 Denition of Limit
LIMIT OF A FUNCTION (INFORMAL DEFINITION)
The notation
lim f x D L
x!c
is read the limit of f(x) as x approaches c is L and means
that the functional values f(x) can be made arbitrarily close
to L by choosi
Vectors
Lines in 3-D
In 3-D, lines which are not parallel may or may not meet. Non-parallel lines which do not meet are said
to be skew.
The vector equation of a line
T
h
e
lines with vector equation r=a +sp and r=b + tq have the same direction if p is a
*strings
Tension on a rope always acts towards the center of the rope
i.
A loose string is said to be slack. There is no tension acting on a slack rope.
ii.
When a rope is being pulled there is tension in the rope
*The difference between a rope and a rode
Momentum and impulse
Momentum
The moment of a body of mass m having a velocity of v is mv. If the units of mass and velocity are kg and
ms-1 respectively, then the units for momentum are newton-seconds (Ns)
Changes in momentum- if the velocity of a body c
Further work on distributions
Probability density function (P.D.F)
A continuous random variable X is given by its probability density function, which is specified for the
range of values for which x is valid. The function can be illustrated by a curve, y=
Inference using normal and t-distribution
Z- Tests
1
2
3
T-tests
This is testing the mean when the population X is normal but the variance 2 is unknown and the
sample size n is small (less than 30)
The t-distribution
The distribution of T is a member of a
Bivariate data
Regression and correlation
Scatter diagrams
Data connecting two variables are known as bivariate data. When pairs
of values are plotted, a scatter diagram is produced.
Dependent and independent variables
If one of the variables has been con
The X2-tests
There are two main situations where an X2 (pronounced kye squared) significance test is used:
An X2 goodness-of-fit-test
This is used when you have some practical data and you want to know how well a particular statistical
distribution, e.g.
1.1a Representations of Function, I
In this lesson, we will review the notion of a function.
Functions are a central object of Calculus. Functions to Calculus are like atoms to chemistry, data to
statistics, or brain to neuroscience. Functions allow us to