Difference Quotient (4 step method of slope)

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Presentation transcript:

Difference Quotient (4 step method of slope) Also known as: (Definition of Limit), and (Increment definition of derivative) f ’(x) = lim f(x+h) – f(x) h→0 h This equation is essentially the old slope equation for a line: x – represents (x1) f (x) – represents (y1) x + h – represents (x2) f (x+h) – represents (y2) f (x+h) – f (x) – represents (y2 – y1) h – represents (x2 – x1) Lim – represents the slope M as h→0

f(x+h) = 3(x2 + 2xh + h2)+ 6(x+h) – 4 given ► combine like terms and organize Notice original f(x) in green remove parentheses substitute (x+h) for every x in f(x) expand (x+h)2 f(x+h) = 3(x+h)2 + 6(x+h) – 4 f(x+h) = 3(x2 + 2xh + h2)+ 6(x+h) – 4 f(x+h) = 3x2 + 6xh + 3h2+ 6x+6h – 4 f(x) = 3 x2 + 6 x – 4 f(x+h) = 3x2 + 6x – 4 + 3h2+ 6xh +6h f(x+h) = 3x2 + 6x – 4 + 3h2+ 6xh +6h

f(x+h) – f(x) = 3h2 + 6xh + 6h h h Note: You should have only “h” terms left in the numerator ►Remove brackets / combine like terms ►Combine numerator and denominator ►Create numerator f(x+h) – f(x) f(x+h) – f(x) = {3x2 + 6x – 4 + 3h2+ 6xh +6h} f(x+h) – f(x) = 3h2+ 6xh +6h – {3x2 + 6x – 4} f(x+h) – f(x) = 3h2 + 6xh + 6h h h

►Cancel h top and bottom ►Factor out common h f(x+h) – f(x) = (3h + 6x + 6) h f(x+h) – f(x) = h(3h + 6x + 6) h h f(x+h) – f(x) = 3h2 + 6xh + 6h h h f(x+h) – f(x) = (3h + 6x + 6) h 1

f’(x) = 6x + 6 f ’(x) = lim f(x+h) – f(x) h→0 h Then If you are evaluating the limit of the equation as h goes to zero f ’(x) = lim f(x+h) – f(x) h→0 h Then Let ‘h’ go to zero f(x+h) – f(x) = h f’(x) = 6x + 6 3h + 6x + 6 3h + 6x + 6 6x + 6 f ’(x) represents the slope of the original equation at any x value.