Barrons AP Calculus

(a)

(b)

(a)

(b)

(c)
(d)

given in the Appendix.

D1. Periodicity and Amplitude

Trigonometric   functions

The trigonometric functions are periodic. A function f is periodic if there is a
positive number p such that f(x + p) = f(x) for each x in the domain of f. The
smallest such p is called the period of f. The graph of f repeats every p units
along the x-axis. The functions sin x, cos x, csc x, and sec x have period 2π; tan x
and cot x have period π.
The function f(x) = A sin bx has amplitude A and period ; g(x) = tan cx has
period .

Example 10 __

Consider the function .

For what    value   of  k   does    f   have    period  2?

What    is  the amplitude   of  f   for this    k?

SOLUTIONS:

Function    f   has period   ;  since   this    must    equal   2,  we  solve   the equation    

,   getting k   =   π.

It  follows that    the amplitude   of  f   that    equals      has a   value   of  .

Example 11 __

Consider the function .

Find (a) the period and (b) the maximum value of f.

What    is  the smallest    positive    x   for which   f   is  a   maximum?

Sketch  the graph.

SOLUTIONS: