Difference quotient algebra

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Transcript of Difference quotient algebra

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Slopes and the Difference QuotientRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is y m = y2 y1 = x2 x1 x Slopes and the Difference Quotient(x1, y1)(x2, y2)Recall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is y m = y2 y1 = x2 x1 x Slopes and the Difference Quotient(x1, y1)(x2, y2)y=y2y1=risex=x2x1=runRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x))

xP=(x, f(x)) Slopes and the Difference Quotienty= f(x) (x1, y1)(x2, y2)y=y2y1=risex=x2x1=runRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) Slopes and the Difference Quotienty= f(x) (x1, y1)(x2, y2)y=y2y1=risex=x2x1=runRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) Slopes and the Difference Quotienty= f(x) (x1, y1)(x2, y2)y=y2y1=risex=x2x1=runhRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), Slopes and the Difference Quotienty= f(x) hRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isSlopes and the Difference Quotienty= f(x) hRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isy m = f(x+h) f(x) = (x+h) x x Slopes and the Difference Quotienty= f(x) hRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isy m = f(x+h) f(x) = (x+h) x x or m = f(x+h) f(x) hSlopes and the Difference Quotienty= f(x) hRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isy m = f(x+h) f(x) = (x+h) x x or m = f(x+h) f(x) hThis is the "difference quotient" formula for slopesSlopes and the Difference Quotienty= f(x) hRecall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isy m = f(x+h) f(x) = (x+h) x x or m = f(x+h) f(x) hf(x+h)f(x) = ybecause f(x+h) f(x) = difference in heightThis is the "difference quotient" formula for slopesSlopes and the Difference Quotienty= f(x) Recall that if (x1, y1) and (x2, y2) are two points then the slope m of the line connecting them is

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) y m = y2 y1 = x2 x1 x Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) be as shown for some y = f(x), then the slope of the cord connecting P and Q (in function notation) isy m = f(x+h) f(x) = (x+h) x x or m = f(x+h) f(x) hh=xf(x+h)f(x) = ybecause f(x+h) f(x) = difference in height andh = (x+h) x = difference in the x's, as shown.This is the "difference quotient" formula for slopes.Slopes and the Difference Quotienty= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient Formulay= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient Formulay= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.Examples of the algebra for manipulating this formula are given below. m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient Formulay= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.Examples of the algebra for manipulating this formula are given below. m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient FormulaExample A. (Quadratics) Given f(x) = x2 2x + 2,f(x+h) f(x) h. simplify its differencequotienty= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.Examples of the algebra for manipulating this formula are given below. m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient FormulaExample A. (Quadratics) Given f(x) = x2 2x + 2,f(x+h) f(x) h= f(x+h) f(x) h. simplify its differencequotienty= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.Examples of the algebra for manipulating this formula are given below. m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient FormulaExample A. (Quadratics) Given f(x) = x2 2x + 2,f(x+h) f(x) h= (x+h)2 2(x+h) + 2 [ x2 2x + 2]hf(x+h) f(x) h. simplify its differencequotienty= f(x) h

xP=(x, f(x)) x+hQ=(x+h, f(x+h)) The goal of simplifying the differencequotient formula is to eliminate the h in the denominator.Examples of the algebra for manipulating this formula are given below. m = f(x+h) f(x) hhf(x+h)f(x) The Algebra of Difference QuotientThe Difference Quotient FormulaExample A. (Quadratics) Given f(x) = x2 2x + 2,f(x+h) f(x) h= (x+h)2 2(x+h) + 2 [ x2 2x + 2]h2xh 2h + h2 h= 2x 2 + h. = f(x+h) f(x) h. simplify its differencequotienty= f(x) http://www.slideshare.net/math123a/4-7polynomial-operationsverticalhThe algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.The Algebra of Difference QuotientExample B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x2The Algebra of Difference QuotientExample B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x2The Algebra of Difference Quotientf(x+h) f(x) h= Example B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x23 (x + h) 23 x2 h The Algebra of Difference Quotientf(x+h) f(x) h= Example B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x23 (x + h) 23 x2 h (3 x h) (3 x) The Algebra of Difference Quotientf(x+h) f(x) h= (3 x h) (3 x) Example B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x23 (x + h) 23 x2 h (3 x h) (3 x) The Algebra of Difference Quotientf(x+h) f(x) h= (3 x h) (3 x) (3 x h) (3 x)Example B. (Rational Functions I) Simplify the difference quotient of f(x) = The algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 x23 (x + h) 23 x2 h (3 x h) (3 x) The Algebra of Difference Quotientf(x+h) f(x) h= (3 x h) (3 x) (3 x h) (3 x)2(3 x) 2(3 x h)h(3 x h) (3 x)= Warning: Its illegal to cancel the ( )s, we have to simplify the numerator,simplifyThe algebra for simplifying the difference quotient of rational functions is the algebra for simplifying complex fractions. To simplify a complex fraction, use the LCD to clear all denominators.3 (x + h) 23 x2 h (3 x h) (3 x) The Algebra of Difference Quotientf(x+h) f(x) h= (3 x h) (3 x) (3 x h) 2(3 x) 2(3 x h)h(3 x h) (3 x)= simplify2hh(3 x h) (3 x)= 2(3 x h) (3 x)= Example B. (Rational Functions I) Simplify the difference quoti