Difference Quotient

Difference Quotient Difference Quotient is an effective tool in calculus. It is used to find out slope of a secant line. Secant line is a line that intersects at any two points on a curve. Difference quotient is used in defining the derivative. Dividing the function difference from the difference of the points is called as difference quotient. . We know that slope is change in y axis over change in x axis. If secant line passes through two points (x, f(x)) and (x + h, f(x + h)). Then the slope of a secant line is calculated as m = (f(x + h) f(x)) / (x + h) x. by simplifying this we get slope = (f(x + h) f(x)) / h and it is denoted by d y/d x. Problem 1: Find the difference quotient of function f(x) = 4x^2 - 1 Solution: Given function isf(x) = 4x^2 -1 = So f(x + h) = 4(x + h) ^2 -1 = Now find, f(x + h) - f(x) = 4(x + h) ^2 -1 (4x^2 -1) = 4(x^2 + h^2 + 2xh) 1 - 4x^2 + 1 = 4x^2 + 4h^2 + 8xh 1 - 4x^2 + 1 = 4h^2 + 8xh = Difference Quotient for function f(x) = (f(x + h) f(x)) / h = (4h^2 + 8xh)/h = 4h + 8x. Problem 2: Find the difference quotient of the function f(x) =3 - 5x Solution: Given function isf(x) = 3 - 5x = So f(x + h) = 3 5(x + h) = Now find, f(x + h) - f(x) = 3 5(x + h) (3 - 5x) = 3 5 x - 5h 3 + 5x = -5h = Difference Quotient for function f(x) = (f(x + h) f(x)) / h = -5h /h = -5.