Find intervals on which the following functions are increasing and decreasing.
1. f(x) = sin 2x : x is greater than or equal to 0 and less than or equal to pi
2. y = sin x + cos x [0,2pi]
2 answers
wouldn't those be the intervals where the derivative is + ?
A function is increasing when the derivative is positive. If you are not studying calculus, then you will have to plot and think about the curves. The sine function rises until the angle (2x in this case) reaches pi/2, and then rises again between 3 pi/2 and 2 pi. That means 0< x< pi/4 and
3 pi/8 < x < pi are the reagions where sin 2x rises. You need to also consider the second period of oscillation when 2x goes from 2 pi to 4 pi. This will give you two more intervals for x.
For the second problem, use calculus or consider using the identity
sin x + cos x =
sqrt2 * sin(x + pi/4)
which reaches a maximum at x = pi/4, as well as 5 pi/4
3 pi/8 < x < pi are the reagions where sin 2x rises. You need to also consider the second period of oscillation when 2x goes from 2 pi to 4 pi. This will give you two more intervals for x.
For the second problem, use calculus or consider using the identity
sin x + cos x =
sqrt2 * sin(x + pi/4)
which reaches a maximum at x = pi/4, as well as 5 pi/4