Some Useful Results from Elementary Calculus

Some Useful Results from Elementary Calculus - (P1 f g = f...

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Some Useful Results from Elementary Calculus Loosely speaking, the time derivative of a function f ( t ) is a new function f' ( t ) that keeps track of the rate of change of f in time. Just as in our formula for velocity, we have, in general: f' ( t ) = Notice that this means we can write: v ( t ) = x' ( t ) . Similarly, we can also take the derivative of the derivative of a function, which yields what is called the second derivative of the original function: f'' ( t ) = We will see later that this enables us to write: a ( t ) = x'' ( t ) , since the acceleration a of an object is equal to the time-derivative of its velocity, i.e. a ( t ) = v' ( t ) . It can be shown, from the above definition for the derivative, that derivatives satisfy certain properties:
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Unformatted text preview: (P1) ( f + g )' = f' + g' • (P2) ( cf )' = cf' , where c is a constant. Without going into more detail about the mathematical nature of derivatives, we will use the following results for the derivatives of some particular functions--given to us courtesy of basic calculus. • (F1) if f ( t ) = t n , where n is a non-zero integer, then f' ( t ) = nt n-1 . • (F2) if f ( t ) = c , where c is a constant, then f' ( t ) = 0 . • (F3a)if f ( t ) = cos wt , where w is a constant, then f' ( t ) = - w sin wt . • (F3b)if f ( t ) = sin wt , then f' ( t ) = w cos wt . These rules, together with (P1) and (P2) above, will give us all the necessary tools to solve many interesting kinematics problems....
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