Difference Quotient

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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 is f(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 is f(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.

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