1. Let f be a twice-differentiable function such that f (2) = 8 and f (4) = 5. Which of the following must be true for the function f on the interval 2 ≤ x ≤ 4.
I. The average value of f is 6.5
II. The average rate of change of f is -1.5
III. The average value of f ' is -1.5
Select one:
a. I, II, and III
b. I and II only
c. II only
d. II and III only
e. III only
the average value of f' on the interval is
∫ f' dx = (f(4)-f(2))/(4-2) = -3/2
Since II and III say the same thing, I pick (D)
we know nothing about the average value of f.
To determine which of the statements are true, we need to use the information given and apply relevant concepts.
I. The average value of f is 6.5:
To find the average value of a function on an interval, you need to calculate the definite integral of the function over the interval and divide it by the width of the interval. In this case, we have the interval 2 ≤ x ≤ 4. The definite integral of function f over this interval is:
∫(2 to 4) f(x) dx = F(4) - F(2)
Since f(2) = 8 and f(4) = 5, the average value of f can be calculated as:
Average value = (F(4) - F(2)) / (4 - 2)
II. The average rate of change of f is -1.5:
The average rate of change of a function over an interval is given by the difference in function values divided by the difference in x-values. Thus, in this case, we need to calculate:
Average rate of change = (f(4) - f(2)) / (4 - 2)
III. The average value of f' is -1.5:
To find the average value of the derivative of a function over an interval, we need to calculate the definite integral of the derivative over the interval and divide it by the width of the interval. In this case, we have the interval 2 ≤ x ≤ 4. The definite integral of f' over this interval is:
∫(2 to 4) f'(x) dx = f(4) - f(2)
Since we know f(2) and f(4), we can calculate the average value of f' as:
Average value of f' = (f(4) - f(2)) / (4 - 2)
Now, let's calculate each of these values to determine which statements are true.
To determine which of the statements are true, we need to analyze the given information about the function and its average values and rates of change.
First, let's find the average value of f on the interval 2 ≤ x ≤ 4. The average value of a function on an interval [a, b] is given by the formula:
Avg(f) = (1 / (b - a)) * ∫(a to b) f(x) dx
Using this formula, we can calculate the average value of f:
Avg(f) = (1 / (4 - 2)) * ∫(2 to 4) f(x) dx
Since we don't have the actual function f, we cannot evaluate the integral to obtain the exact average value. Therefore, we cannot determine whether statement I is true or false.
Next, let's find the average rate of change of f on the interval 2 ≤ x ≤ 4. The average rate of change of a function on an interval [a, b] is given by the formula:
Avg rate of change(f) = (f(b) - f(a)) / (b - a)
Using this formula, we can calculate the average rate of change of f:
Avg rate of change(f) = (f(4) - f(2)) / (4 - 2)
= (5 - 8) / 2
= -3 / 2
= -1.5
So, statement II is true. The average rate of change of f on the interval 2 ≤ x ≤ 4 is indeed -1.5.
Finally, let's find the average value of f' (the derivative of f) on the interval 2 ≤ x ≤ 4. The average value of a derivative on an interval [a, b] is given by the formula:
Avg(f') = (1 / (b - a)) * ∫(a to b) f'(x) dx
We do not have any information about the derivative of f, so we cannot evaluate the integral to obtain the exact average value. Therefore, we cannot determine whether statement III is true or false.
In conclusion, the only statement that can be determined to be true is statement II. Therefore, the correct answer is c. II only.