Difference Quotient Examples

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Difference quotient examples help in finding derivative of a function. Dividing the function difference from the difference of the points is called as difference quotient. This gives slope of a secant line passing through two points. The formula for finding difference quotient is (f(x + h) – f(x)) / h and it is denoted by dy /dx.

Problem 1: Consider a function f(x) = 2x^2 – 5 and x changes from 1 to 1.4. Find the value of the difference quotient in this situation.

Solution: Given function is f(x) = 2x^2 – 5

=> So f(x + h) = 2(x + h) ^2 – 5

=> Difference Quotient for function f(x) = (f(x + h) – f(x)) / h 

= (2(x + h) ^2 – 5 – (2x^2 – 5)) / h= (2x^2 + 4xh + 2h^2 – 5 -2x^2 + 5)/ h

= (4xh + 2h^2)/ h = 4x + 2h

=> Given x changes from 1 to 1.4

=> So dx = 1.4 – 1 = 0.4. (Hence, h = 0.4)

=> Using x = 1 and h = 0.4 then

=> Difference quotient = 4x + 2h = 4(1) + 2(0.4)) = 4 + 0.8 = 4.08

=> Therefore, 4.08 is the slope of secant line when x changes from 1 to 1.3


Problem 2: Consider the function f(x) = 7x – 2. Find the difference quotient and find dy when dx = 2.

Solution: Given function is f(x) = 7x – 2

=> So f(x + h) = 7(x +h) – 2

=> Difference Quotient for function f(x) = (f(x + h) – f(x)) / h

= (7(x +h) – 2 – (7x – 2)) / h = 7

=> Difference quotient is always 7 for this function.

=> We have dy / dx = 7 

=> dy = 7 * dx = 7 * 2 = 14.

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