Limits and Derivatives eni
Limits and Derivativ
es
The Idea of Limits
The Idea of
2
x 4
Limits
Consider the function f ( x )
x
f(x)
1.9 1.99 1.999 1.9999
2
x 2
2.0001 2.001 2.01 2.1
The Idea of
Limits
Consider the function
x 1.9 1.99 1.999 1.9999
f(x) 3.9 3.99 3.999 3.9999
2
x 4
f ( x)
x 2
2
2.0001 2.001 2.01 2.1
un4.0001 4.001 4.01 4.1
defined
The Idea of
Limits
Consider the function
x
g ( x) x 2
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
g ( x) x 2
y
2
x
O
If a function f(x) is a continuous
f ( x) f ( x0 )
at x0, lim
then
.
x x0
approaches to,
but not equal to
lim f ( x) 4
x 2
lim g ( x) 4
x 2
The Idea of
Limits
Consider the function
x
g(x)
-4
-3
-2
-1
h( x )
0
1
x
x
2
3
4
The Idea of
Limits
Consider the function
x -4
h(x) -1
-3
-1
-2
-1
-1
-1
h( x )
0
undefined
1
1
x
x
2
2
3
3
4
4
lim h( x) 1
x 0
lim h( x)
x 0
lim h( x) 1
x 0
does not
exist.
A function f(x) has limit l at x0 if f
(x) can be made as close to l as
we please by taking x sufficientl
y close to (but not equal to) x0.
We write
lim f ( x) l
x x0
Theorems On Limits
Theorems On Limits
Theorems On Limits
Theorems On Limits
Limits at Infinity
Limits at Infinity
1
Consider f ( x) 2
x 1
Generalized, if
lim f ( x)
x
then
k
lim
0
x f ( x )
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
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 Cu
rve
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
Slope of AT lim
x 0
y
f ( x x) f ( x)
lim
x
0
x
x
provided that the limit exists.
Increments
The increment △x of a variable is
the change in x from a fixed value
x = x0 to another value x = x1.
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
+ △x) – f(x0).
Derivatives
(A) Definition of Derivative.
The derivative of a function y = f(x)
with respect to x is defined as
y
f ( x x) f ( x)
lim
lim
x 0 x
x 0
x
provided that the limit exists.
The derivative of a function y = f(x) wit
h respect to x is usually denoted by
dy d
,
f (x),
dx dx
y' ,
f ' ( x).
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
vative of the function with respect to x e
xists at x = x0.
The value of the derivative of y
= f(x) with respect to x at x = x0 i
s denoted
dy
by dx
x x0
or
f ' ( x0 ).
To obtain the derivative
of a function by its defi
nition is called different
iation of the function fr
om first principles.
Contoh Soal
Jika diketahui , carilah
Jawab
Carilah kemudian carilah
Rumus-Rumus Diferensial
Contoh - contoh
2.
3.
4.
5.
6.
7.
8.
misal
9.
Soal Latihan
es
The Idea of Limits
The Idea of
2
x 4
Limits
Consider the function f ( x )
x
f(x)
1.9 1.99 1.999 1.9999
2
x 2
2.0001 2.001 2.01 2.1
The Idea of
Limits
Consider the function
x 1.9 1.99 1.999 1.9999
f(x) 3.9 3.99 3.999 3.9999
2
x 4
f ( x)
x 2
2
2.0001 2.001 2.01 2.1
un4.0001 4.001 4.01 4.1
defined
The Idea of
Limits
Consider the function
x
g ( x) x 2
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
g ( x) x 2
y
2
x
O
If a function f(x) is a continuous
f ( x) f ( x0 )
at x0, lim
then
.
x x0
approaches to,
but not equal to
lim f ( x) 4
x 2
lim g ( x) 4
x 2
The Idea of
Limits
Consider the function
x
g(x)
-4
-3
-2
-1
h( x )
0
1
x
x
2
3
4
The Idea of
Limits
Consider the function
x -4
h(x) -1
-3
-1
-2
-1
-1
-1
h( x )
0
undefined
1
1
x
x
2
2
3
3
4
4
lim h( x) 1
x 0
lim h( x)
x 0
lim h( x) 1
x 0
does not
exist.
A function f(x) has limit l at x0 if f
(x) can be made as close to l as
we please by taking x sufficientl
y close to (but not equal to) x0.
We write
lim f ( x) l
x x0
Theorems On Limits
Theorems On Limits
Theorems On Limits
Theorems On Limits
Limits at Infinity
Limits at Infinity
1
Consider f ( x) 2
x 1
Generalized, if
lim f ( x)
x
then
k
lim
0
x f ( x )
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
Theorems of Limits at Infinity
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 Cu
rve
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
Slope of AT lim
x 0
y
f ( x x) f ( x)
lim
x
0
x
x
provided that the limit exists.
Increments
The increment △x of a variable is
the change in x from a fixed value
x = x0 to another value x = x1.
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
+ △x) – f(x0).
Derivatives
(A) Definition of Derivative.
The derivative of a function y = f(x)
with respect to x is defined as
y
f ( x x) f ( x)
lim
lim
x 0 x
x 0
x
provided that the limit exists.
The derivative of a function y = f(x) wit
h respect to x is usually denoted by
dy d
,
f (x),
dx dx
y' ,
f ' ( x).
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
vative of the function with respect to x e
xists at x = x0.
The value of the derivative of y
= f(x) with respect to x at x = x0 i
s denoted
dy
by dx
x x0
or
f ' ( x0 ).
To obtain the derivative
of a function by its defi
nition is called different
iation of the function fr
om first principles.
Contoh Soal
Jika diketahui , carilah
Jawab
Carilah kemudian carilah
Rumus-Rumus Diferensial
Contoh - contoh
2.
3.
4.
5.
6.
7.
8.
misal
9.
Soal Latihan