Difference Quotient - CSUSM
Difference Quotient
Difference Quotient (4 step method of slope)
Also known as: (Definition of Limit), and (Increment definition of derivative)
f '(x) = lim f(x+h) ? f(x)
h0
h
This equation is essentially the old slope equation for a line:
m = y2 - y1 x2 - x1
f (x+h)
? represents (y2)
f (x)
? represents (y1)
x
? represents (x1)
x + h
- represents (x2)
h
? represents the change in x or (x2 ? x1) or x
f (x+h) ? f (x)
? represents (y2 ? y1)
Lim
? represents the slope M as h0
M = y2 - y1 = f (x + h) - f (x) = f (x + h) - f (x)
x2 - x1
(x + h) - x
h
f(x+h)
f(x) = 2(x ? 4)2 + 8 Secant lines Tangent = ?1.5(x?3) + 8.2
f(x) h
x
x+h
As `h' gets smaller, the value of (x+h) gets closer to (x) and thus f(x+h) gets closer to f(x), and the slope of the secant line gets closer to the slope of the tangent line at (x). And so as h0, we get the limit of the equation at (x)
James S Jun 2010 r6
Difference Quotient
The Difference Quotient is an algebraic approach to the Derivative ( dy ) and is sometimes referred to as the dx
"Four Step Method." It is a way to find the slope of a line tangent to some function f(x) at some point (x) on the function that is continuous at that (x).
The idea of a limit is to get very close to a given value of (x) in f(x), even if f(x) is not defined at (x) and so in our equation, h0 (h approaches zero), but does not necessarily equal zero.
Process:
f(x) = 3x2 + 6x ? 4
given
Step 1: Substitute (x + h) into f(x)
f(x+h) = 3(x+h)2 + 6(x+h) ? 4 f(x+h) = 3(x2 + 2xh + h2)+ 6(x+h) ? 4 f(x+h) = 3x2 + 6xh + 3h2 + 6x + 6h ? 4 f(x+h) = [3x2 + 6x ? 4] + 3h2 + 6xh + 6h
substitute (x+h) for every x in f(x) expand remove parentheses combine like terms and organize;
Notice original f(x) in [bracket]
Step 2: Organize terms of the Numerator ( f(x+h) ? f(x) )
[f(x+h)] ? [f(x)] = ([3x2 + 6x ? 4] + 3h2 + 6xh + 6h) ? [3x2 + 6x ? 4] assembled numerator portion
[f(x+h)] ? [f(x)] = 3h2 + 6xh + 6h
combine like terms
Step 3: Organize Difference Quotient (numerator/denominator)
f(x+h) ? f(x) = 3h2 + 6xh + 6h
h
h
organize difference quotient
f(x+h) ? f(x) = h(3h + 6x + 6) = 3h + 6x + 6
h
h
1
factor out common "h"
Note: You should always be able to factor out a common `h'
Step 4: Evaluate the Limit of the Quotient
Evaluate:
Lim f(x+h) ? f(x) = 3h + 6x + 6
h0
h
1
0
as h0
Lim f(x+h) ? f(x) = 6x + 6 = 6x + 6
h
1
Lim as h0
The slope (M) of the line tangent to f(x) = 6x + 6 at any given (x)
James S Jun 2010 r6
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- slope of line tangent to curve
- 1 a numerical scheme edu
- difference quotient csusm
- the notebook project ap calculus ab
- math 1190 Œexam 1 version 1 solutions
- ap calculus njctl
- secant ti83 dec 9 06
- the di erence quotient purdue university
- ti 84 graphing calculator guide designed to accompany
- 2 limitsand derivatives t 2 1 the tangent andvelocity problems
Related searches
- find the difference quotient solver
- quotient and remainder calculator
- quotient in long division
- estimate the quotient 486 divided by 5
- divide give quotient and remainder
- quotient and remainder calculator polynomials
- calculator with quotient and remainder
- derivative quotient calculator
- quotient rule calculator with steps
- quotient rule simplify calculator
- quotient rule calculator calculus
- find the quotient and simplify