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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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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
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 - 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 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.
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