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

Limits and Derivatives


The idea of limits

The Idea of Limits


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


Limits and derivatives

does not exist.


Limits and derivatives

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 limits1

Theorems On Limits


Theorems on limits2

Theorems On Limits


Theorems on limits3

Theorems On Limits


Limits at infinity

Limits at Infinity


Limits and derivatives

Limits at Infinity

Consider


Limits and derivatives

Generalized, if

then


Limits and derivatives

Theorems of Limits at Infinity


Limits and derivatives

Theorems of Limits at Infinity


Limits and derivatives

Theorems of Limits at Infinity


Limits and derivatives

Theorems of Limits at Infinity


Contoh contoh

Contoh - contoh

Contoh 1

Contoh 2

Bila f(x) = 13

Contoh 3


Limits and derivatives

Contoh 4

Contoh 5

=(6)(1)=6


Limits and derivatives

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 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.


Limits and derivatives

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


Limits and derivatives

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 .


Limits and derivatives

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


Contoh soal

ContohSoal

Jika diketahui, carilah

Jawab

Carilahkemudiancarilah


Rumus rumus diferensial

Rumus-RumusDiferensial


Contoh contoh1

Contoh - contoh

2.

3.


Limits and derivatives

4.

5.

6.


Limits and derivatives

7.

misal

8.


Limits and derivatives

9.


Soal latihan

SoalLatihan


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