5. y is a function of x , and the relation y = x 2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y . y = x 2
6. Since the value of y depends on a given value of x , we call y the dependent variable and x the independent variable and of the function y = x 2 .
22. Consider the function The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x )
23. Consider the function The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f ( x ) 3.9 3.99 3.999 3.9999 un-defined 4.0001 4.001 4.01 4.1
24. Consider the function The Idea of Limits x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 g ( x ) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 x y O 2
25. If a function f ( x ) is a continuous at x 0 , then . approaches to, but not equal to
29. A function f ( x ) has limit l at x 0 if f ( x ) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x 0 . We write
42. 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.
43. Increments The increment △ x of a variable is the change in x from a fixed value x = x 0 to another value x = x 1 .
44. For any function y = f ( x ), if the variable x is given an increment △ x from x = x 0 , then the value of y would change to f ( x 0 + △ x ) accordingly. Hence thee is a corresponding increment of y (△ y ) such that △ y = f ( x 0 + △ x ) – f ( x 0 ) .
45. Derivatives (A) Definition of Derivative. The derivative of a function y = f ( x ) with respect to x is defined as provided that the limit exists.
46. The derivative of a function y = f ( x ) with respect to x is usually denoted by
47. 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 = x 0 if the derivative of the function with respect to x exists at x = x 0 .
48. The value of the derivative of y = f ( x ) with respect to x at x = x 0 is denoted by or .
49. To obtain the derivative of a function by its definition is called differentiation of the function from first principles .
