Limits and derivatives
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Limits and Derivatives. The Idea of Limits. The Idea of Limits. Consider the function. The Idea of Limits. Consider the function. y. 2. x. O. The Idea of Limits. Consider the function. If a function f ( x ) is a continuous at x 0 , then . .

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Consider the function

The Idea of Limits

Consider the function


Consider the function1

The Idea of Limits

Consider the function


Consider the function2

y

2

x

O

The Idea of Limits

Consider the function


If a function f x is a continuous at x 0 then

If a function f(x) is a continuous at x0, then .

approaches to, but not equal to


Consider the function3

The Idea of Limits

Consider the function


Consider the function4

The Idea of Limits

Consider the function



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 sufficiently close to (but not equal to) x0. We write













Contoh contoh
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 curve1
The Slope of the Tangent to a Curve

The slope of the tangent to a curve y = f(x) with respect to x is defined as

provided that the limit exists.


Increments

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 is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence there is a corresponding increment of y(△y) such that △y = f(x0 + △x) – f(x0).


Derivatives

Derivatives

The derivative of a function y = f(x) with respect to x is defined as

provided that the limit exists.

(A) Definition of Derivative.


The derivative of a function y f x with respect to x is usually denoted by

The derivative of a function y = f(x) with respect to x is usually denoted by


The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to xexists at x = x0.


The value of the derivative of y f x with respect to x at x x 0 is denoted by or

The value of the derivative of y = f(x) with respect to x at x = x0 is denoted by or .


To obtain the derivative of a function by its definition is called differentiation of the function from first principles.


Contoh soal
Contoh called Soal

Jika diketahui, carilah

Jawab

Carilahkemudiancarilah


Rumus rumus diferensial
Rumus-Rumus called Diferensial


Contoh contoh1
Contoh called - contoh

2.

3.


4. called

5.

6.


7. called

misal

8.


9. called


Soal latihan
Soal called Latihan


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