Log in Page Discussion History Go to the site toolbox

Guide to Calculus Derivatives

From BluWiki

Calculus evolved from several problems facing mathematics during the seventeeth century. One such problem was to find a line that is tangent to a curve at a specific point. While this is easily geometrically defined for circles (the line touching the circle at exactly one point P) it does not provide a general solution for all curves. The problem boils down to finding the slope of the tangent at point P.

Definition of a Derivative

By understanding the limit, it is easy to see that the two points (x,f(x)) and (x+h,f(x+h)) will become infinitely close as <math>h->0</math>, because h is becoming infinitely small. Slope is given as the change in <math>y</math> over the change in <math>x</math>. The change in <math>y</math> is <math>f(x+h)-f(x)</math>. The change in <math>x</math> is <math>(x+h)-x</math>, or <math>h</math>.


Derivatives do not exist in certain situations. If at a value of x there is a hole, discontinuity, jump, asymptote, vertical tangent, or a sharp turn in the graph a derivative does not exist at that point.

HOLES: A function with a "hole" at x=7
DISCONTINUITIES: A function with a defined point at x=7
JUMPS: A greatest integer function with multiple jumps
ASYMPTOTES: A function with an asymptote at x=7
VERTICAL TANGENTS: The slope of a vertical tangent is undefined
SHARP TURNS: One sided limits of the slope do not agree


Site Toolbox:

Personal tools
GNU Free Documentation License 1.2
This page was last modified on 20 October 2005, at 17:38.
Disclaimers - About BluWiki