Find the average rate of change of the function over the given interval. You must reduce and simplify your answer, if possible.
a. f(x)=cot x, [pi/6, pi/2]
b. f(theta)= -2 + cos (theta), [-pi, 0]
I'll tackle the first one. The second one will be left for you as an exercise.
f(x)=cot x, [pi/6, pi/2]
To find the average rate of change over the interval, we divide the definite integral by the interval, namely:
Δf(x)/Δx
=∫f(x)dx / (π/2-π/6)
=∫cot(x)dx / (π/3)
=[ln(sin(x))] /(π/3)
=[0-ln(sin(π/6)]/(π/3)
=ln(2)/π/3
=3ln(2)/π
To find the average rate of change of a function over a given interval, follow these steps:
1. Calculate the difference in function values between the endpoints of the interval.
2. Divide the difference by the difference in the independent variable (x or theta).
a. Average rate of change of f(x) = cot x over [pi/6, pi/2]:
Step 1: Calculate the difference in function values:
f(pi/2) - f(pi/6)
Applying the cotangent function:
cot(pi/2) - cot(pi/6)
Step 2: Calculate the difference in the independent variable:
(pi/2) - (pi/6)
Now, we have both the difference in function values and the difference in the independent variable. Let's simplify the expression:
cot(pi/2) is undefined because it represents a vertical asymptote.
Using the identity cot(x) = 1/tan(x):
cot(pi/2) = 1/tan(pi/2)
tan(pi/2) is also undefined, so we can't get a numeric value for cot(pi/2). However, we can still find the average rate of change by simplifying the expression further:
1/tan(pi/2) = cos(pi/2) / sin(pi/2)
cos(pi/2) = 0 and sin(pi/2) = 1, so the expression becomes:
0/1 = 0
Therefore, the average rate of change of f(x) = cot x over [pi/6, pi/2] is 0.
b. Average rate of change of f(theta) = -2 + cos(theta) over [-pi, 0]:
Step 1: Calculate the difference in function values:
f(0) - f(-pi)
Substituting the values:
(-2 + cos(0)) - (-2 + cos(-pi))
Step 2: Calculate the difference in the independent variable:
0 - (-pi) = pi
Now, let's simplify the expression:
(-2 + cos(0)) is equal to -2 + 1 = -1
(-2 + cos(-pi)) is equal to -2 + (-1) = -3
Therefore, the average rate of change of f(theta) = -2 + cos(theta) over [-pi, 0] is:
(-1 - (-3)) / pi = (3 - 1) / pi = 2 / pi