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What is the meaning of continuous first derivatives of a function?
A continuous function is one where there are no discontinuities or step changes in the function, i.e. for a small change in input value, as that small change approaches zero, there is a progressively smaller change in output value. There are many definitions, some formal and some intuitive, for continuous functions. The definition given above is intuitive. The same definition can be give to the deriviatives or the integrals of a function. Continousness does not depend on being a deriviative or integral.
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How do you calculate instantaneouse speed?
The first derivative of position calculates instantaneous speed. If you do not know how to differentiate, then either use a calculator, use slope on the graph (if it's linear or piecewise-linear), or learn how... The derivative according to Newton is the limit as h approaches 0 of (f(x+h)-f(x))/h, and according to Leibniz it's the limit as x approaches a of (f(x)-f(a))/(x-a).
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