If f is a function such that
(f(b) - f(a))/(b-a) =2, then which of the following statements must be true
a) f(a) = f(b) =2
b) the slope of the tangent line to the function at x =a is 2
c) The average rate of change of the function on the interval [a,b] is 2
d) the linear approximation for
f(x) at x =a is y= 2
looks like (c) to me. That fraction is ∆y/∆x, right?
b) would be true if one were dealing with the lim(f) as b >>>a. But that is not true.
c. average is a powerful term. True
.
Correct answer is C
To determine which of the statements must be true, let's consider the given equation:
(f(b) - f(a))/(b - a) = 2
This equation represents the average rate of change of the function f(x) on the interval [a, b]. It tells us that for every 1 unit increase in x (from a to b), the corresponding change in f(x) is 2 units.
Now, let's analyze each statement and see if it aligns with the given equation:
a) f(a) = f(b) = 2:
This statement does not necessarily have to be true. The equation only tells us the average rate of change, so it does not guarantee that f(a) and f(b) are equal to 2. There can be various values for f(a) and f(b) as long as the average rate of change is 2.
b) The slope of the tangent line to the function at x = a is 2:
This statement also does not necessarily have to be true. The given equation represents the average rate of change over the interval, but it does not tell us specifically about the slope of the tangent line at x = a. So, the slope of the tangent line can be different from 2.
c) The average rate of change of the function on the interval [a, b] is 2:
This statement must be true because the given equation directly represents the average rate of change. Therefore, the average rate of change on the interval [a, b] is indeed 2.
d) The linear approximation for f(x) at x = a is y = 2:
This statement does not necessarily have to be true either. The given equation does not provide information about the exact function values or its linear approximation. Therefore, we cannot conclude that the linear approximation for f(x) at x = a is y = 2.
In summary, the statement that must be true based on the given equation is c) The average rate of change of the function on the interval [a, b] is 2.