Consider the function
The Idea of
Limits
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x)
Consider the function
The Idea of
Limits
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
f(x) 3.9 3.99 3.999 3.9999
un-defined 4.0001 4.001 4.01 4.1
Consider the function
The Idea of
Limits
x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1
g(x) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1
x y O 2
2
)
(
x
x
g
2 )
(x x
If a function
f
(
x
) is a continuous
at x
0, then
.
approaches to, but not equal to
)
(
)
(
lim
0 0x
f
x
f
xx
4
)
(
lim
2
f
x
x
4
)
(
lim
2
g
x
Consider the function
The Idea of
Limits
x -4 -3 -2 -1 0 1 2 3 4
g(x)
x
x
x
Consider the function
The Idea of
Limits
x -4 -3 -2 -1 0 1 2 3 4
h(x) -1 -1 -1 -1
un-defined 1 2 3 4
x
x
x
does not
exist.
1
)
(
lim
0
h
x
x
1
)
(
lim
0
h
x
x
)
(
lim
0
h
x
A function
f
(
x
) has limit
l
at
x
0if
f
(
x
) can be made as close to
l
as
we please by taking
x
sufficientl
y close to (but not equal to)
x
0.
We write
l
x
f
x
Limits at Infinity
Consider
1
1
)
(
2
x
x
Generalized, if
then
(
)
lim
f
x
x
0
)
(
lim
f
x
Contoh - contoh
Contoh 1 Contoh 2
Bila f(x) = 13
Contoh 3
Contoh 4
Contoh 5
=(6)(1)=6
Contoh 6 Contoh 7
The Slope of the Tangent to a Curv
e
The slope of the tangent to a curve
y
=
f
(
x
) with respect to
x
is defined
as
provided that the limit exists.
Increments
The increment △
x
of a variable is
the change in
x
from a fixed value
For any function y = f(x), if the variable x i s given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence there is a correspondi ng increment of y(△y) such that △y = f(x0
Derivatives
(A) Definition of Derivative.
The derivative of a function
y
=
f
(
x
)
with respect to
x
is defined as
provided that the limit exists.
The derivative of a function
y
=
f
(
x
) wit
h respect to x is usually denoted by
,
dx
dy
),
(
x
f
dx
d
y
,'
).
(
The process of finding the derivative of a function is called differentiation.
A function y = f(x) is said to be differenti able with respect to x at x = x0 if the deri
The value of the derivative of
y
=
f
(
x
) with respect to
x
at
x
=
x
0i
s denoted
by or .
0 x x
dx
dy
)
(
To obtain the derivative
of a function by its defi
nition is called
different
iation
of the function
fr
Contoh Soal
Jika diketahui , carilah Jawab
Carilah kemudian carilah
Contoh - contoh
2.
3.
4.
5.
6.
7.
misal
8.
9.
