Showing posts with label simplifying difference quotient. Show all posts
Showing posts with label simplifying difference quotient. Show all posts

Monday, August 13

Difference (Newton) Quotient Made Simple


To set up a difference quotients for a given function requires an understanding of a function notation. Given the function f(x)=4x^2-3x-7. This notation is read as “f of x equals..” This implies that the value of the function, that is the y-value depends upon the replacement for x. We get the numerical value for the function by substituting a number for ‘x’. If a non-numerical quantity is substituted for ‘x’, we get an expression rather than a numerical value. One important point to be remembered is careful use of parenthesis which is essential. For instance, f(x)=4x^2-3x-7; f(3)=4(3)^2 – 3(3)-7=36-9-7=20

Derivative Quotients  at x for a function f is given by, [f(x+h)-f(x)]/h. Sometimes it is written using delta(x) for the change in x and delta(y) for change in y; delta(x)=h and delta(y)=f(x+delta(x))- f(x). The Difference Quotient is so called as it involves the operations subtraction and division. The common forms of Difference Quotient are as follows:
1. [f(x+h) – f(x)]/h
2. [f(a+h) – f(a)]/h
3. [f(x+delta(x)) – f(x)]/delta(x)

Simplifying Difference Quotient
The difference quotient is simplified to get a h or delta(x) in the denominator which can be canceled to get the final value. Let us consider a difference quotient example to understand the step involved in simplifying difference quotient ; f(x) = 4x^2-3x-7. First we need to find the function f(x+h),which we can get by substituting (x+h) in all x in the given function.  f(x+h)= 4(x+h)^2 – 3(x+h) – 7 = 4(x^2+2xh+h^2) – 3x – 3h – 7= 4x^2+8xh + 4h^2-3x – 3h – 7. Subsituting f(x+h) in the difference quotient, we get,
Difference Quotient = [f(x+h) – f(x)]/h
       = {[4x^2+8xh+4h^2-3x-3h-7] – [4x^2-3x-7]}/h
        = 4x^2+8xh+4h^2-3x-3h-7 -4x^2+3x+7]/h  (opening the parenthesis)
On simplification, we get
        = [8xh+4h^2-3h]/h  
        = h[8x +4h-3]/h            (taking h common)
        = (8x +4h-3)     (canceling h)

Difference Quotient Example
Given function, f(x) = 2x^2-1
First we need to calculate f(x+h) which is got by substituting (x+h) in all x of the function
f(x+h) = 2(x+h)^2 -1 = 2(x^2+2xh+h^2) -1 = 2x^2+ 4xh+2h^2 -1
Next we substitute f(x+h) and f(x) in difference quotient
 [f(x+h) – f(x)]/h ={[2x^2 +4xh +2h^2-1]- [2x^2 -1]}/h
= [2x^2+4xh +2h^2-2x^2+1]/h
= [4xh+2h^2]/h  (combining like terms)
= h[4x +2h]/h      (taking h common)
= [4x +2h] (canceling h)

Newton Quotient
The difference quotient is attributed to Sir Isaac Newton and hence given the name Newton Quotient. The slope of a line through the points [(x+h),f(x+h)] and [x, f(x)] is given by [f(x+h) – f(x)]/h. This expression is the Newton Quotient or Newton’s difference quotient.
Newton Quotient  =[f(x+h) – f(x)]/h