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Limits and Derivatives

Limits and Derivatives. Concept of a Function. y = x 2. 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 .

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Limits and Derivatives

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  1. Limits and Derivatives

  2. Concept of a Function

  3. y = x2 y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.

  4. 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 = x2.

  5. Notation for a Function : f(x)

  6. The Idea of Limits

  7. The Idea of Limits Consider the function

  8. The Idea of Limits Consider the function

  9. y 2 x O The Idea of Limits Consider the function

  10. If a function f(x) is a continuous at x0, then . approaches to, but not equal to

  11. The Idea of Limits Consider the function

  12. The Idea of Limits Consider the function

  13. does not exist.

  14. 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

  15. Theorems On Limits

  16. Theorems On Limits

  17. Theorems On Limits

  18. Theorems On Limits

  19. Exercise 12.1 P.7

  20. Limits at Infinity

  21. Limits at Infinity Consider

  22. Generalized, if then

  23. Theorems of Limits at Infinity

  24. Theorems of Limits at Infinity

  25. Theorems of Limits at Infinity

  26. Theorems of Limits at Infinity

  27. Exercise 12.2 P.13

  28. Theorem where θ is measured in radians. All angles in calculus are measured in radians.

  29. Exercise 12.3 P.16

  30. The Slope of the Tangent to a Curve

  31. 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.

  32. Exercise 12.4 P.18

  33. Increments The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.

  34. 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 thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) – f(x0).

  35. 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.

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

  37. 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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