Numerical+representations+-+Difference+quotients

Difference Quotients ** media type="custom" key="5778629" Hi, my name’s Borat. I like you, I like math. In Kazakhstan school they teach me – how you call? – difference quotients! High Five! I teach you.
 * Numerical Representations-

The difference quotient is the average slope of a function between two points, when these points are represented as: ( x, f(x)) And (x+h , f(x+h)) According to the formula for slope - m = (y2 - y1) / (x2 - x1)

This becomes: m = [ f (x + h) – f(x) ] / [ (x + h) – x ] Which simplifies to: m = [ f (x + h) – f(x) ] / h

__Example 1:__ Find the difference quotient of function f defined by

__Solution to Example 1__
 * f(x) = 2x + 5**

> > f(x + h) = 2(x + h) + 5 > > > [ f (x + h) – f(x) ] / h = [ 2(x + h) + 5 – (2 x + 5) ] / h > == 2 h / h= = 2
 * We first need to calculate f(x + h).
 * We now substitute f(x + h) and f(x) in the definition of the difference quotient by their expressions
 * We simplify the above expression.
 * The answer is 2.

Very Nice!

In this case, the h value does not matter. Whatever the h, or delta-x is, the slope will always be two. Why ? Because the function is a straight line with slope of 2!

Wowawewa!

Now we try another one, where the h matters!

__Example 2: __ Find the difference quotient of the following function **f(x) = 2x 2 + x - 2 ** __Solution to Example 2 __ > > f(x + h) = 2(x + h) 2 + (x + h) - 2 > > [ f (x + h) – f(x) ] / h = > > [ 2(x + h) 2 + (x + h) - 2 - ( 2x 2 + x - 2 ) ] / h > =[ 4 x h + 2 h 2 + h] / h= 4 x + 2 h + 1 For this one now, plug in different h and you will get the slope from x to x plus h!
 * We first calculate f(x + h). 
 * We now substitute f(x + h) and f(x) in the difference quotient 
 * We expand the expressions in the numerator and group like terms.

Try this video, which shows how to find a simple difference quotient:

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** f’(x) = lim x--> c [f(x) – f(c)]/ (x-c) ** The slope (instantaneous rate of change) of the secant line can be calculated using the points A and B. The cordinates of A (x, f(x)) and B ( f(x+h)) can be used to get: slope = [ f (x + h) – f(x) ] / [ (x + h) – x ] or slope = [ f (x + h) – f(x) ] / h ** Example: ** Find f’(5) if f(x) = x^2 -3x-4. lim x à 5 [f(x) – f(5)]/(x-5) = lim x à 5 (x^2 – 3x-4-6)/ (x-5) = lim x à 5 (x^2 – 3x-10)/ (x-5) = lim x à  5 (x-5)( (x+2)/ (x-5) = x+2  lim x à 5 (x+2) = 7  **Note**: this is the slope of the tangent line or the instantaneous rate of change at f(5 )
 * Difference Quotient: **** A derivative at a point **** (derivative at x = c form) **
 * Note ** : slopes of secant lines (in the graph below, the line passing through the points A and B is a secant line) are equal to the average rates of change, the slopes of the tangent lines are the instantaneous rates of change

As a check, you may plug in //y = [f(x) – f(5)] / (x-5)//or //y = (x^2 – 3x – 10 )/ (x-5)// into your graphing calculator in order to view the limit.

As you can see from the chart provided, there is a removable discontinuity at x =5, but also that y approaches the limit 7 as x approaches 5.
 * **x** || **quotient** ||
 * 4.996 || 6.996 ||
 * 4.997 || 6.997 ||
 * 4.998 || 6.998 ||
 * 4.999 || 6.999 ||
 * 5 || ERROR ||
 * 5.001 || 7.001 ||
 * 5.002 || 7.002 ||
 * 5.003 || 7.003 ||

Sources: [] [] //Calculus Concepts and Application//, Paul A. Foerster.

Topic: Numerical Representations Group Members: Suchana and Reuben

|| 9.64 || 9.27 || 4.36 || 4.36 || 9 || 36.63 || 7 || 6 || 4 || 4 || 6 || 27 || 8.32 || 7.635 ||  4.18 ||  4.18 ||  7.5 ||  31.815 ||
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Your mathematical notation needs improvement. It’s a difficult to follow and at times isn’t appropriate. You shouldn’t use ^ for example. Try using an equation editor. You also don’t talk about the limit of the difference quotient as the derivative. Lacking thoroughness. I like the use of other media and theme. Quality is lacking.

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