Applications of the calculus depend on interpretations of the derivative, such as the slope of a graph, a velocity or an acceleration, marginal profit or cost or revenue, a rate of growth or of decay. For example, acceleration is the derivative of speed for a moving vehicle. Thus if the acceleration of an object is known to be constantly 7, then its speed s(t) as a function of time satisfies the equation s’(t) = 7. This is called a differential equation: it is an equation involving the derivative of a function. The solution to this equation is not a number; it is the function s(t) = 7t + C, where 7 is the constant acceleration and C is the number s(0), the value of the speed at the time coordinate 0. In general, in differential equations the unknowns do not stand for numbers but for functions, and the solutions are functions.
KeywordsTaylor Series Series Solution Decimal Place Solution Function Remainder Term
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