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## PowerPoint Slideshow about 'Limits and Derivatives' - andra

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

### Increments

### Derivatives

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

### 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.

approaches to, but not equal to

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

Limits at Infinity

Consider

Generalized, if

then

Contoh 6

Contoh 7

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.

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).

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 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.

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