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Stitz-Zeager_College_Algebra_e-book

# How do we know this again 9 this is all of course a

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Unformatted text preview: l name. Definition 10.3. Periodic Functions: A function f is said to be periodic if there is a real number c so that f (t + c) = f (t) for all real numbers t in the domain of f . The smallest positive number p for which f (t + p) = f (t) for all real numbers t in the domain of f , if it exists, is called the period of f . We have already seen a family of periodic functions in Section 2.1: the constant functions. However, we leave it to the reader as an exercise to show that, despite being periodic, constant functions have no period. Returning to the circular functions, we see that by Deﬁnition 10.3, f (t) = cos(t) is periodic, since cos(t + 2π · k ) = cos(t) for any integer k . To determine the period of f , we need to ﬁnd the smallest real number p so that f (t + p) = f (t) for all real numbers t or, said diﬀerently, the smallest positive real number p such that cos(t + p) = cos(t) for all real numbers t. We know that cos(t + 2π ) = cos(t) for all real numbers t but the question remains if any smaller real number will do the trick. Suppose p > 0 and cos(t + p) = cos(t) for...
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