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- 1. Limits and Derivatives
- 2. Concept of a Function
- 3. FUNCTIONS <ul><li>“ FUNCTION” indicates a relationship among objects. </li></ul><ul><li>A FUNCTION provides a model to describe a system. </li></ul><ul><li>A FUNCTION expresses the relationship of one variable or a group of variables (called the domain) with another variables( called the range) by associating every member in the domain to a unique member in range. </li></ul>
- 4. TYPES OF FUNCTIONS <ul><li>LINEAR FUNCTIONS </li></ul><ul><li>INVERSE FUNCTIONS </li></ul><ul><li>EXPONENTIAL FUNCTIONS </li></ul><ul><li>LOGARITHMIC FUNCTIONS </li></ul>
- 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 .
- 10. Notation for a Function : f ( x )
- 21. The Idea of Limits
- 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
- 26. Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 g ( x )
- 27. Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 h ( x ) -1 -1 -1 -1 un-defined 1 2 3 4
- 28. does not exist.
- 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
- 30. Theorems On Limits
- 31. Theorems On Limits
- 32. Theorems On Limits
- 33. Theorems On Limits
- 34. Limits at Infinity
- 35. Limits at Infinity Consider
- 36. Generalized, if then
- 37. Theorems of Limits at Infinity
- 38. Theorems of Limits at Infinity
- 39. Theorems of Limits at Infinity
- 40. Theorems of Limits at Infinity
- 41. The Slope of the Tangent to a Curve
- 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 .
- 50. Differentiation Rules 1.
- 51. Differentiation Rules 1.
- 52. Differentiation Rules 2.
- 53. Differentiation Rules 2.
- 54. Differentiation Rules 2.
- 55. Differentiation Rules 3.
- 56. Differentiation Rules 3.
- 57. Differentiation Rules 4. for any positive integer n
- 58. Differentiation Rules 4. for any positive integer n Binominal Theorem
- 59. Differentiation Rules 5. product rule
- 60. Differentiation Rules 5. product rule
- 62. Differentiation Rules 6. where v ≠ 0 quotient rule
- 63. Differentiation Rules 6. where v ≠ 0 quotient rule
- 65. Differentiation Rules 7. for any integer n
- 68. DIFFERENTIATION RULES <ul><li>y,u and v are functions of x. a,b,c, and n are constants (numbers). </li></ul>The derivative of a constant is zero. Duh! If everything is constant, that means its rate, its derivative, will be zero. The graph of a constant, a number is a horizontal line. y=c. The slope is zero. The derivative of x is 1. Yes. The graph of x is a line. The slope of y = x is 1. If the graph of y = cx, then the slope, the derivative is c.
- 69. MORE RULES <ul><li>When you take the derivative of x raised to a power (integer or fractional), you multiply expression by the exponent and subtract one from the exponent to form the new exponent. </li></ul>Example:
- 70. OPERATIONS OF DERIVATIVES <ul><li>The derivative of the sum or difference of the functions is merely the derivative of the first plus/minus the derivative of the second. </li></ul><ul><li>The derivative of a product is simply the first times the derivative of the second plus second times the derivative of the first. </li></ul><ul><li>The derivative of a quotient is the bottom times the derivative of the top, minus top times the derivative of the bottom….. All over bottom square.. </li></ul><ul><li>TRICK: LO-DEHI – HI-DELO </li></ul><ul><ul><ul><ul><ul><li>LO 2 </li></ul></ul></ul></ul></ul>
- 71. JUST GENERAL RULES <ul><li>If you have constant multiplying a function, then the derivative is the constant times the derivative. See example below: </li></ul><ul><li>The coefficient of the x 6 term is 5 (original constant) times 7 (power rule.) </li></ul>
- 72. SECOND DERIVATIVES <ul><li>You can take derivatives of the derivative. Given function f(x), the first derivative is f’(x). The second derivative is f’’(x), and so on and so forth. </li></ul><ul><li>Using Leibniz notation of dy/dx </li></ul>For math ponders , if you are interesting in the Leibniz notation of derivatives further, please see my article on that. Thank you. Hare Krishna >=) –Krsna Dhenu
- 73. EXAMPLE 4: <ul><li>Find the derivative: </li></ul><ul><li>Use the power rule and the rule of adding derivatives. </li></ul><ul><li>Note 3/2 – 1 = ½. x ½ is the square root of x. </li></ul><ul><li>Easy eh?? </li></ul>
- 74. EXAMPLE 5 <ul><li>Find the equation of the line tangent to y = x 3 +5x 2 –x + 3 at x=0. </li></ul><ul><li>First find the (x,y) coordinates when x = 0. When you plug 0 in for x, you will see that y = 3. (0,3) is the point at x=0. </li></ul><ul><li>Now, get the derivative of the function. Notice how the power rule works. Notice the addition and subtraction of derivative. Notice that the derivative of x is 1, and the derivative of 3, a constant, is zero. </li></ul>
- 75. EX 5 (continued) <ul><li>Now find the slope at x=0, by plugging in 0 for the x in the derivative expression. The slope is -1 since f’(0) = -1. </li></ul><ul><li>Now apply it to the equation of a line. </li></ul>
- 76. EX 5. (continued) <ul><li>Now, plug the x and y coordinate for x 0 and y 0 respectively. Plug the slope found in for m. </li></ul><ul><li>And simplify </li></ul><ul><li>On the AP, you can leave your answer as the first form. (point-slope form) </li></ul>
- 77. EXAMPLE 6 <ul><li>Find all the derivatives of y = 8x 5 . </li></ul><ul><li>Just use the power rule over and over again until you get the derivative to be zero. </li></ul><ul><li>See how the power rule and derivative notation works? </li></ul>

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