# Total Distance Traveled by a Particle

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Last updated 7 years ago

Discipline:
Math
Subject:
Calculus   Total Distance Traveled by a ParticleBy: Bradley Allen

History of DerivativesDerivatives were first discovered around the 1630's and 1640's by Roberval and nearly at the same time, Fermat used maxima and the infinitesimal to find the tangent to a curve. Some people give credit to Fermat with discovering derivatives, but it wasn't until Leibniz and Newton defined a method of tangents that became accepted. The key that earned Leibniz and Newton credit was creating the fundamental theorem of calculus.http://www.math.wpi.edu/IQP/BVCalcHist/calc2.htmlTotal DistanceThere are a few ways to find total distance of f(x). If f(x) is given, simply graph f(x) on a number line and find f '(x). Next, find the points where the derivative changes signs. Then take the absolute value of the second sign change or ending point and subtract the first sign change or starting point. Continue doing this method until the whole graph is completed. Then add up all the individual distances to get the total distance. Another option if f(x) is given is to take the derivative of f(x) and set it equal to 0. Then take the second derivative and find what that equals when f '(x) = 0. This will determine if a local maximum or minimum is present. Split the original interval into multiple parts based on the local extema and add up all the distances for these intervals to get the total distance.Another option is if f(x) is not given but f '(x) is, find the points where the derivative changes signs. Then integrate f '(x) for the intergrals from the starting point or the x-value of the derivative's sign change to the ending point or the next x-value of the derivative's sign change. Continue this method until the whole graph is completed. Once each of these distances has been calculated, add them up to find the total distance.