diff quotient-6 - THE DIFFERENCE QUOTIENT I The ability to...

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THE DIFFERENCE QUOTIENT I. The ability to set up and simplify difference quotients is essential for calculus students. It is from the difference quotient that the elementary formulas for derivatives are developed. II. Setting up a difference quotient for a given function requires an understanding of function notation. III. Given the function: f(x) = 2 3x 4x 5 A. This notation is read “f of x equals . . .”. B. The implication is that the value of the function (the y-value) depends upon the replacement for “x”. C. If a number is substituted for “x”, a numerical value for the function is found. D. If a non-numerical quantity is substituted for “x”, an expression is found rather than a numerical value. E. Careful use of parentheses is essential! IV. Examples using f(x) = 2 3x 4x 5 A. 2 f(4) 3(4) 4(4) 5 27 = = B. 2 f( 3) 3( 3) 4( 3) 5 34 = = C. f(a) = 2 3a 4a 5 D. 2 f(2a 3) 3(2a 3) 4(2a 3) 5 = 2 3(4a 12a 9) 8a 12 5 = + + = 2 2 12a 36a 27 8a 12 5 12a 44a 34 + + = + E. = 2 f(x h) 3(x h) 4(x h) 5 + = + + 2 2 3(x 2xh h ) 4x 4h 5 + + 2 2 3x 6xh 3h 4x 4h 5 = + + F. = 2 f(5 h) 3(5 h) 4(5 h) 5 + = +
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