Transcript EE005_fhs_lnt_001_Sep09 - EE005-Calculus-UCSI

CHAPTER 1 :
Introduction
Limits and Continuity
What is Calculus?
 Mathematics of motion and change
 Whenever there is motion or growth, whenever there is
variable forces at work
 Learning calculus is not the same as learning arithmetic,
algebra, and geometry. In those subjects, you learn
primarily how to calculate with numbers, how to simplify
algebraic expressions and calculate with variables, and how
to reason with points, lines, and figures in the plane.
Calculus involves those techniques and skills but develops
others as well, with greater precision and at deeper level.
Learning Calculus
 Read the text
 Do the homework
 Use calculators and computers
 Try writing your own notes and short descriptions
 Rewrite definitions etc to increases persistency in
understanding
Limits
Overview
 One of the chapters that connects Calculus and Algebra and Trigo.
 Straightforward calculations and can be solved by simple
substitution.
 Limits – describe the way a function changes
 Some functions vary continuously, small changes in x produce
only small changes in f(x). Other functions can have values that
jump or vary erratically.
Speed
 A rate of change
 A moving body’s average speed over any particular
time interval is the amount of distance covered during
the interval divided by the length of the interval.
Example:
A rock falls of a tall cliff. What is its average speed
during :(i) First 2 seconds (t = 2 sec) of fall
(ii) At time t = 2 seconds
Gradient of secants as approaching limit
Average Rate of Change
y f ( x2 )  f ( x1 )

x
x2  x1
f(x)
f ( x1  h)  f ( x1 )

,h  0
h
f(x2)
∆y
f(x1)
∆x
x1
x2
Limits
 f(x) approaches arbitrarily closer to a h1 when x approaches h2
x 1
f ( x) 
x 1
( x  1)  ( x  1)

,x 1
x 1
 x 1
2
lim f ( x)  2, x  1
x 1
or
x 2 1
lim 
 2, x  1
x 1
x 1
Informal Limits
 f(x), x=x0, x0≠x0
 If f(x) close to L for all x values close to x0
 f(x) approaches limit L as x approaches x0
lim f ( x)  L
x  x0
 “informal” because phrases like arbitrarily close or
sufficiently close are imprecise
Rules for Finding Limits
 Limits can be assessed algebraically, using arithmetic and rules
Theorem 1: Limit Rules
The following rules hold if lim f ( x)  L and lim g ( x)  M
x c
x c
where L, M, c and k are real numbers
Theorem 1 : Limit Rules
1. Sum Rule:
lim  f ( x)  g ( x)  L  M
lim  f ( x)  g ( x)  L  M
lim  f ( x)  g ( x)  L  M
lim k  f ( x)  kL (for any number k)
x c
2. Difference Rule
x c
3. Product Rule
x c
4. Constant Multiple Rule
x c
5. Quotient Rule
lim
x c
6. Power Rule
f ( x) L
 ,M  0
g ( x) M
If M and n are integers, then
lim  f ( x)
x c
m
n
provided L is a real number
m
n
L
m
n
Theorem 2
Limits of Polynomials can be found by substitution
If P( x)  a n x n  a n 1 x n 1  ...  a 0 ,
then
n
n 1
P
(
x
)

P
(
c
)

a
c

a
c
 ...  a0
n
n 1
lim
x c
Theorem 3
Limits of Rational Functions can be found by substitution if the limit of the
denominator is not zero
If P(x) and q(x) are polynomials and Q(c) is not zero, then
P ( x ) P (c )

lim
Q (c )
x c Q ( x )
Other methods include creating and canceling a common factor.
Solving Limit Questions
THREE step/method rule
STEP 1: Substitute/Replace x value
If unsuccessful
STEP 2: Factorise to cancel nominator
If unsucceful
STEP 3: Conjugation
REMEMBER  There is no such thing as an undefined limit!
0 or any other number is still a limit