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Limits and derivatives

by Laxmikant Deshmukh, Lecturer at Shri Sant Gajanan Maharaj College of Engineering Shegaon on Nov 02, 2011

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Limits and derivativesPresentation Transcript

• Limits and Derivatives
• Concept of a Function
• FUNCTIONS
• “ FUNCTION” indicates a relationship among objects.
• A FUNCTION provides a model to describe a system.
• 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.
• TYPES OF FUNCTIONS
• LINEAR FUNCTIONS
• INVERSE FUNCTIONS
• EXPONENTIAL FUNCTIONS
• LOGARITHMIC FUNCTIONS
• 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
• 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 .
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• Notation for a Function : f ( x )
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• The Idea of Limits
• 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 )
• 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
• 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
• If a function f ( x ) is a continuous at x 0 , then . approaches to, but not equal to
• Consider the function The Idea of Limits x -4 -3 -2 -1 0 1 2 3 4 g ( x )
• 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
• does not exist.
• 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
• Theorems On Limits
• Theorems On Limits
• Theorems On Limits
• Theorems On Limits
• Limits at Infinity
• Limits at Infinity Consider
• Generalized, if then
• Theorems of Limits at Infinity
• Theorems of Limits at Infinity
• Theorems of Limits at Infinity
• Theorems of Limits at Infinity
• The Slope of the Tangent to a Curve
• 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 = x 0 to another value x = x 1 .
• 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 ) .
• 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.
• 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 = x 0 if the derivative of the function with respect to x exists at x = x 0 .
• The value of the derivative of y = f ( x ) with respect to x at x = x 0 is denoted by or .
• To obtain the derivative of a function by its definition is called differentiation of the function from first principles .
• Differentiation Rules 1.
• Differentiation Rules 1.
• Differentiation Rules 2.
• Differentiation Rules 2.
• Differentiation Rules 2.
• Differentiation Rules 3.
• Differentiation Rules 3.
• Differentiation Rules 4. for any positive integer n
• Differentiation Rules 4. for any positive integer n Binominal Theorem
• Differentiation Rules 5. product rule
• Differentiation Rules 5. product rule
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• Differentiation Rules 6. where v ≠ 0 quotient rule
• Differentiation Rules 6. where v ≠ 0 quotient rule
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• Differentiation Rules 7. for any integer n
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• DIFFERENTIATION RULES
• y,u and v are functions of x. a,b,c, and n are constants (numbers).
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.
• MORE RULES
• 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.
Example:
• OPERATIONS OF DERIVATIVES
• The derivative of the sum or difference of the functions is merely the derivative of the first plus/minus the derivative of the second.
• The derivative of a product is simply the first times the derivative of the second plus second times the derivative of the first.
• 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..
• TRICK: LO-DEHI – HI-DELO
• LO 2
• JUST GENERAL RULES
• If you have constant multiplying a function, then the derivative is the constant times the derivative. See example below:
• The coefficient of the x 6 term is 5 (original constant) times 7 (power rule.)
• SECOND DERIVATIVES
• 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.
• Using Leibniz notation of dy/dx
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
• EXAMPLE 4:
• Find the derivative:
• Use the power rule and the rule of adding derivatives.
• Note 3/2 – 1 = ½. x ½ is the square root of x.
• Easy eh??
• EXAMPLE 5
• Find the equation of the line tangent to y = x 3 +5x 2 –x + 3 at x=0.
• 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.
• 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.
• EX 5 (continued)
• 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.
• Now apply it to the equation of a line.
• EX 5. (continued)
• Now, plug the x and y coordinate for x 0 and y 0 respectively. Plug the slope found in for m.
• And simplify
• On the AP, you can leave your answer as the first form. (point-slope form)
• EXAMPLE 6
• Find all the derivatives of y = 8x 5 .
• Just use the power rule over and over again until you get the derivative to be zero.
• See how the power rule and derivative notation works?