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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 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

slide10
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

slide23
Contoh 4

Contoh 5

=(6)(1)=6

slide24
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.

slide28
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.

slide31
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.
slide33

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

slide38
4.

5.

6.

slide39
7.

misal

8.

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