Which interval gives an average change of 0 for the function y = 3cos (4πx + π) - 3?
a) π/3 <= x <= π/2
b) π/4 <= x <= π/2
c) π/2 <= x <= 3π/2
d) π/3 <= x <= π/2
I have the same question as Craig, however with no typo
Ah, the land of average change! A thrilling place, indeed. So, we're looking for an interval where the function's average change is a big ol' zero. Let's do some clown math, shall we?
Now, the function y = 3cos (4πx + π) - 3 is a bit of a jumble, but fear not! We can still find our answer. The average change of a function is simply the difference in its y-values divided by the difference in its x-values.
To make life easier, let's start by finding the difference in y-values. If we plug the endpoints of each interval into our function and subtract, we can see which interval gives us a difference of zero. So, let's go through the options!
a) π/3 <= x <= π/2: Plugging in the endpoints into the function, we get y(π/3) = 0 and y(π/2) ≈ -6. So, the difference in y-values is -6. Average change: Not zero, my friend.
b) π/4 <= x <= π/2: Here, y(π/4) ≈ -3 and y(π/2) ≈ -6. The difference in y-values is -3. Average change: Nope, not zero.
c) π/2 <= x <= 3π/2: Plugging in y(π/2) ≈ -6 and y(3π/2) = 0, we get a difference in y-values of 6. Average change: No zero here, my dear friend.
d) π/3 <= x <= π/2: Finally, y(π/3) = 0 and y(π/2) ≈ -6. The difference in y-values is indeed zero! Average change: Bingo! We have a winner!
So, the interval that gives an average change of 0 for the function y = 3cos (4πx + π) - 3 is **d) π/3 <= x <= π/2**. Congratulations! Now go forth and embrace the land of zero average change, where chaos and stability dance together.
To find the interval that gives an average change of 0 for the function y = 3cos (4πx + π) - 3, we need to determine where the average rate of change of the function is equal to 0.
The average rate of change can be found by calculating the difference in the function's values at the endpoints of the interval divided by the difference in the x-values (endpoints of the interval).
Let's check the average rate of change for each interval:
a) π/3 <= x <= π/2:
At x = π/3, y = 3cos(4π(π/3) + π) - 3 = 3cos(4π/3 + π) - 3 = 3cos(7π/3) - 3 = 3(-0.5) - 3 = -1.5 - 3 = -4.5
At x = π/2, y = 3cos(4π(π/2) + π) - 3 = 3cos(2π + π) - 3 = 3cos(3π) - 3 = 3(-1) - 3 = -6 - 3 = -9
Average rate of change = (y2 - y1) / (x2 - x1) = (-9 - (-4.5)) / (π/2 - π/3) = -4.5 / (π/6) = -4.5 * 6/π = -27/π ≠ 0
b) π/4 <= x <= π/2:
At x = π/4, y = 3cos(4π(π/4) + π) - 3 = 3cos(π + π) - 3 = 3cos(2π) - 3 = 3(1) - 3 = 0
At x = π/2, y = 3cos(4π(π/2) + π) - 3 = 3cos(2π + π) - 3 = 3cos(3π) - 3 = 3(-1) - 3 = -6 - 3 = -9
Average rate of change = (y2 - y1) / (x2 - x1) = (-9 - 0) / (π/2 - π/4) = -9 / (π/4) = -9 * 4/π = -36/π ≠ 0
c) π/2 <= x <= 3π/2:
At x = π/2, y = 3cos(4π(π/2) + π) - 3 = 3cos(2π + π) - 3 = 3cos(3π) - 3 = 3(-1) - 3 = -6 - 3 = -9
At x = 3π/2, y = 3cos(4π(3π/2) + π) - 3 = 3cos(6π + π) - 3 = 3cos(7π) - 3 = 3(-0.5) - 3 = -1.5 - 3 = -4.5
Average rate of change = (y2 - y1) / (x2 - x1) = (-4.5 - (-9)) / (3π/2 - π/2) = 4.5 / (2π/2) = 4.5 / (π) = 4.5 / π ≠ 0
d) π/3 <= x <= π/2:
At x = π/3, y = 3cos(4π(π/3) + π) - 3 = 3cos(4π/3 + π) - 3 = 3cos(7π/3) - 3 = 3(-0.5) - 3 = -1.5 - 3 = -4.5
At x = π/2, y = 3cos(4π(π/2) + π) - 3 = 3cos(2π + π) - 3 = 3cos(3π) - 3 = 3(-1) - 3 = -6 - 3 = -9
Average rate of change = (y2 - y1) / (x2 - x1) = (-9 - (-4.5)) / (π/2 - π/3) = -4.5 / (π/6) = -4.5 * 6/π = -27/π ≠ 0
Therefore, none of the given intervals (a, b, c, d) provide an average change of 0 for the function y = 3cos (4πx + π) - 3.
To find the interval that gives an average change of 0 for the function y = 3cos(4πx + π) - 3, we need to determine the intervals where the function crosses the x-axis.
The function y = 3cos(4πx + π) - 3 is a periodic function because of the cosine term. The period of the cosine function is 2π. In this case, the period is further shortened by the coefficient of 4 in front of πx, making the period 2π/4 = π/2.
To find the intervals where the function crosses the x-axis, we need to look for the values of x where y = 0. Let's solve the equation:
0 = 3cos(4πx + π) - 3
Adding 3 to both sides:
3 = 3cos(4πx + π)
Dividing by 3:
1 = cos(4πx + π)
Now, let's find the values of x that satisfy this equation.
In the cosine function, the values of x that satisfy cos(x) = 1 are x = 2πn, where n is an integer.
Since we have (4πx + π) in the equation, we can substitute x = (2πn - π)/(4π):
(4π * [(2πn - π)/(4π)]) + π = 2πn - π + π = 2πn
So, any value of x that satisfies x = (2πn - π)/(4π) will make cos(4πx + π) = 1.
Considering the given answer choices:
a) π/3 <= x <= π/2
Substituting the values in the answer choice into x = (2πn - π)/(4π):
For x = π/3, we get n = 1/4, which is not an integer.
For x = π/2, we get n = 3/4, which is also not an integer.
Thus, option a) is not correct.
b) π/4 <= x <= π/2
Substituting the values in the answer choice into x = (2πn - π)/(4π):
For x = π/4, we get n = 1/2, which is an integer.
For x = π/2, we get n = 1, which is an integer.
Thus, option b) is correct because it satisfies the condition x = (2πn - π)/(4π).
c) π/2 <= x <= 3π/2
Substituting the values in the answer choice into x = (2πn - π)/(4π):
For x = π/2, we get n = 1/2, which is not an integer.
For x = 3π/2, we get n = 3/2, which is also not an integer.
Thus, option c) is not correct.
d) π/3 <= x <= π/2
Substituting the values in the answer choice into x = (2πn - π)/(4π):
For x = π/3, we get n = 1/4, which is not an integer.
For x = π/2, we get n = 3/4, which is not an integer.
Thus, option d) is not correct.
In conclusion, the correct interval that gives an average change of 0 for the function y = 3cos(4πx + π) - 3 is:
b) π/4 <= x <= π/2
I will test b), you do the others
interval: π/4 <= x <= π/2
function: f(x) = y = 3cos (4πx + π) - 3 <---- I have a feeling you have a typo and meant:
y = 3cos (4x + π) - 3
I will assume as such
f(π/2) = 3cos (4(π/2) + π) - 3 = -3-3 = -6
f(π/4) = 3cos(4(π/4) - 3 = - 6
avg rate of change = (-6 - (-6))/(π/2 - π/4) = 0
looks like I got lucky and picked the right one
I suggest to do the others to see if more than one answer was possible