When the graph of y equal to 3cos2x achieves minimum what is the value of the y coordinate
In point where firs derivative = 0
function have loca minimum or maximum.
If second derivative < 0 that is local maxsimum.
If second derivative > 0 that is local minimum.
If expression 3cos2x mean :
3 cos (2 x )
then first derivative =
- 6 sin ( 2 x )
- 6 sin( 2 x ) = 0 when
sin ( 2 x ) = 0
sin theta = 0 when theta = 0
in this case 2 x = 0
when x = 0 or x = pi / 2
The period of sin x is 2 n pi
The period of sin ( 2 x ) is n pi
where n is some intefer number
So - 6 sin( 2 x ) = 0 when
x = n pi + 0 = n pi
or
x = n pi + pi / 2
or
x = n pi - pi / 2
Second derivative = - 12 cos ( 2 x )
For x = n pi second derivative < 0
for that's values of x function have maximum
For x = n pi + pi / 2 second derivative > 0
and
For x = n pi - pi / 2 second derivative > 0
for that's values of x function also have minimum
So function 3 cos (2 x ) have local minimums when
x = n pi + pi / 2
and
x = n pi - pi / 2
Given y = 3cos2x
you know it has amplitude of 3, so its minimum value is y = -3, since there is no y-translation.
Unless you are specifically doing calculus, I'd surely use my knowledge of trig to answer this one.
Good analysis, though, Bosnian!
To find the value of the y-coordinate when the graph of y = 3cos(2x) achieves its minimum, we need to understand the behavior of the cosine function.
The cosine function oscillates between a maximum value of 1 and a minimum value of -1. The number inside the cosine function, in this case 2x, determines the period of the oscillation.
In the given equation y = 3cos(2x), the coefficient in front of x is 2, which means the period of the oscillation is halved compared to the usual cosine function.
To find when the graph achieves its minimum, we need to find the value of x for which the cosine function reaches its minimum of -1.
In the standard cosine graph, the minimum occurs at an angle of π radians or 180 degrees. However, in our equation y = 3cos(2x), the period is halved, so the minimum will occur at 180/2 = π/2 radians or 90 degrees.
To find the value of the y-coordinate at this x-value, we plug π/2 into the equation:
y = 3cos(2(π/2))
= 3cos(π)
= 3(-1)
= -3
Therefore, when the graph of y = 3cos(2x) achieves its minimum, the value of the y-coordinate is -3.