When you are looking at a graph, a minimum point would be where the curve is decreasing, then begins to increase. Right at the point where it switches, the slope is a horizontal line, or 0. We can take the derivative is f(x), then look for all the x values where the slope (which is equal to the first derivative) is equal to zero.
f'(x) = 2 * -4sin(2x - pi)
The 2 comes from the derivative of the inside, 2x-pi.
So now set the derivative equal to 0.
-8sin(2x-pi) = 0
We can drop the -8 by dividing both sides by -8.
sin(2x-pi) = 0
This can be rewritten as arcsin(0) = 2x-pi
So when theta equals 0, what is the value of sin(theta)? At an angle of 0, there is just a horizontal line pointing to the right on the unit circle with length of 1. Sine is y/h, but there is no y value so it is just 0. If arcsin(0) = 0, we can now set 2x-pi = 0
2x = pi
x = pi/2
This is a critical number. To find the minimum value between 0 and pi, we need to find the y values for the endpoints and the critical number.
f(0) = -4
f(pi/2) = 4
f(pi) = -4
So the minimum points are at x=0 and x=pi
63+74+x=180
180-63-74=43
x=43
Answer:
a=45 b=65
Step-by-step explanation:
You know that all the angle measures of a triangle add to 180. So the equation would be angle a + angle b + 70 = 180. Subsitute and simplify and you get 16x -2 + 70 = 180. Subtract 70 from 180 to get 16x -2=110 add 2 to get 16x =110. Divide each side by 16 to gte x=7. Subsitute x = 7 into the equations for angle a and angle b to get the answers above
Answer:
A ≈ 183 cm²
Step-by-step explanation:
Step 1: Find the area of the 2 circles
A = πr²
r = 6/2 = 3
A = π(3)²
A = 9π
Since there are <em>2</em> circles, 9π(2) = 18π
Step 2: Find area of <em>entire </em>rectangle
A = lw
A = 12(20)
A = 240
Step 3: Subtract the area of the circles from the area of the entire rectangle
240 - 18π
Shaded area = 183.451 cm²
And we have our answer!
