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
so if you think about, the problem presents us with a function, f, and 2 different values of x that are put into f, which are a and b.
if f(b) and f(a) yield y values, and b and a are x values, doesn't that make the given formula the slope formula? see:
f(b) = y2, while f(a) = y1 --> b = x2, and a = x1, so the given formula becomes
y2 - y1 / x2 - x1 = 2
So some slope of f is equal to 2. This can narrow down your answer to either b or c. But if you choose b, you'll see that this doesn't account for x=b. Choosing c, you can understand that the slope is equal to 2 where [a,b]; the average slope (or rate of change) in the interval [a,b] is 2
The correct statement must be c. The average rate of change of the function on the interval [a, b] is 2.
We can see that the given equation [(f(b)-f(a))/(b-a)]=2 represents the average rate of change of the function f(x) on the interval [a, b]. This equation gives us the slope of the secant line passing through (a, f(a)) and (b, f(b)).
Option a, f(a) = f(b) = 2, does not necessarily hold true. The equation [(f(b)-f(a))/(b-a)]=2 does not provide any information about the specific values of f(a) or f(b).
Option b, the slope of the tangent line to the function at x = a is 2, is not necessarily true. The given equation represents the average rate of change, not the instantaneous rate of change (slope of the tangent line at any point).
Option d, the linear approximation for f(x) at x = a is y = 2, is not necessarily true. The equation [(f(b)-f(a))/(b-a)]=2 only provides information about the average rate of change and does not give any information about the specific function values.
Therefore, the correct statement is c. The average rate of change of the function on the interval [a, b] is 2.
To find the correct answer, let's break down the given information.
The expression [(f(b) - f(a))/(b - a)] is the formula for the average rate of change of the function f(x) on the interval [a, b]. It represents the slope of the line connecting the points (a, f(a)) and (b, f(b)).
Given that [(f(b) - f(a))/(b - a)] = 2, this means that the average rate of change of the function f(x) on the interval [a, b] is 2.
Now let's analyze each of the answer choices to determine which one is true:
a. f(a) = f(b) = 2: We cannot infer this from the given information. The fact that the average rate of change is 2 does not tell us anything about the specific values of f(a) and f(b).
b. The slope of the tangent line to the function at x = a is 2: This statement does not necessarily have to be true. Although the average rate of change is 2, the slope of the tangent line to the function at x = a can be different. The average rate of change represents the overall trend over the interval [a, b] and does not guarantee the same slope at every point in that interval.
c. The average rate of change of the function on the interval [a, b] is 2: This statement is already given in the question. It is true and matches the information provided. Therefore, this statement must be true.
d. The linear approximation for f(x) at x = a is y = 2: We cannot determine whether this statement is true or not based solely on the average rate of change being 2. The linear approximation for f(x) at x = a would depend on additional information, such as the derivative of the function at x = a.
Therefore, the correct answer is c. The average rate of change of the function on the interval [a, b] is 2.