local+linearity

Sabrinbrin: Local Linearity
 * function has to be:
 * differentiable
 * continuous
 * one can zoom in enough to make the function appear as a perfectly straight line (redundant?)
 * A function will not have local linearity at- no matter how far you zoom in, it will never look like a straight line
 * cusp
 * [[image:http://www.sosmath.com/calculus/diff/der01/der01_2.gif width="114" height="80" caption="http://www.sosmath.com/calculus/diff/der01/der01_2.gif"]]
 * corner point
 * [[image:http://www.mathwarehouse.com/algebra/linear_equation/images/absolute-value-graphs/absolute-value-diagram.gif width="119" height="105" caption="http://www.mathwarehouse.com/algebra/linear_equation/images/absolute-value-graphs/absolute-value-diagram.gif"]]
 * jump discontinuities (at the point of discontinuity
 * [[image:http://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png width="104" height="99" caption="http://upload.wikimedia.org/wikipedia/commons/e/e6/Discontinuity_jump.eps.png"]]

Genevieve Tabby: The equation of local linearization derives from the point-slope form:  //y// – //y//1 //= m//(//x// – //x//1) where m=slope and  (x1, y1) is the point on the line you are trying to find.  We can say that f( x1)= y1 and the m= f ' (x1). So, if we put this information in the point-slope form, we get this:

y-f(x1)= f ' (x)(x-x1) y= f(x) + f '(x)(x-x1) L(x) = f(x) + f '(x)(x-x1) OR ∆y= f'(x)∆x where ∆y is the change in y and ∆x is the change in x. ∆y is called a differential.

So, for example: Find a linear approximation to estimate the value of (3.02)⁴. So, f(x) = x⁴ ; x=3 f(3)= 81 f '(x) = 4x³ ;f '(3)=108 So, using ∆y= f ' (x)∆x: y-81=108(x-3) y= 81+108(3.02-3) y= 81+108(.02)

__y= 83.16__

Marlee Madora Examples of these graphs are (Ms. Sweeney: I am assuming that you are giving examples of graphs for which there is a point of nondifferentiability, or in other words a place where they are NOT locally linear!) cusp: y=x^2/3  corner point: any absolute value function-y=lxl

discontinuity: many piece wise functions, or a function such as x^2+5x+6/(x+2) which simplifies into (x+3)(x+2)/(x+2) since the x+2's cross out, -2 will give you a removable discontinuity

Zach Fusfeld: In addition to what marlee has already said about the points on a graph at which the function is nondifferentiable, i feel it is important to mention that while there can be removable discontinuities, asymptotes in equations such as y = 1/x are also undifferentiable at certain points. in the case of y = 1/x, the equation is undifferentiable at x = 0 Ian Wilson

Piecewise functions can also result in an extant point that is non-differential, as indeed with an absolute value graph. In point of fact I do not believe that a piecewise graph could possibly have less than one point that was not either locally linear, or a removable discontinuity, assuming it cannot be also written as an ordinary function in some manner. But perhaps my imagination has forgotten something.

I may also be wrong about this, but I do believe that ∆y is not properly called a differential, that, in my mind, being the limit as ∆y approaches 0.But I would be splitting hairs to say so in any case.

Sam Ostrum: I thought this might add something...

It is a graph of x^2... and as the pictures progress the graph gets more and more zoomed in. In the last picture, one can see that the graph is locally linear.