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Guide to Calculus Derivatives

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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

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This page was last modified on 20 October 2005, at 17:38.
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