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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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 x0, then .

approaches to, but not equal to


The Idea of Limits

Consider the function


The Idea of Limits

Consider the function


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


Theorems On Limits


Theorems On Limits


Theorems On Limits


Theorems On Limits


Limits at Infinity


Limits at Infinity

Consider


Generalized, if

then


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 Curve


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

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

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 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 = x0 is denoted by or .


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


ContohSoal

Jika diketahui, carilah

Jawab

Carilahkemudiancarilah


Rumus-RumusDiferensial


Contoh - contoh

2.

3.


4.

5.

6.


7.

misal

8.


9.


SoalLatihan


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