AP Calculus AB Exam Question:

Let f(x) be a differentiable function defined on the interval [a, b]. The graph of f(x) is given below:

(a) Use the graph to find the exact value of f'(c), where c is a real number in the interval (a, b).

(b) Given that f(x) satisfies the conditions of the Fundamental Theorem of Calculus, use your answer from part (a) to evaluate the following definite integral:

∫︎[a, b] f'(x) dx

Answer:

(a) To find the value of f'(c), we can use the relationship between the derivative of a function and the slope of its graph at a point. From the graph, we can observe that the slope of the graph is positive for values of x greater than c, and negative for values of x less than c. Therefore, the value of f'(c) is 0.

Explanation:

The slope of a function at a given point is equal to the derivative of the function at that point. In this case, we are interested in finding the derivative at the point where x = c. By observing the graph, we can see that for values of x greater than c, the graph is increasing, indicating a positive derivative. Similarly, for values of x less than c, the graph is decreasing, indicating a negative derivative. At the point where x = c, the graph does not have a steep incline or decline, which suggests that the slope of the graph (and therefore the derivative) is 0 at that point. Hence, f'(c) = 0.

(b) From the Fundamental Theorem of Calculus, we know that if f(x) is a differentiable function defined on the interval [a, b], then the definite integral of the derivative of f(x) from a to b is equal to f(b) - f(a):

∫︎[a, b] f'(x) dx = f(b) - f(a)

Since the given function f(x) satisfies the conditions of the Fundamental Theorem of Calculus, we can evaluate the definite integral using the given information.

In part (a), we found that f'(c) = 0. Therefore, according to the Fundamental Theorem of Calculus:

∫︎[a, b] f'(x) dx = f(b) - f(a)

Substituting f'(c) = 0, we get:

∫︎[a, b] f'(x) dx = f(b) - f(a) = f(b)

Since the graph of f(x) is given, we can determine the value of f(b) by finding the y-coordinate where the graph intersects the line x = b. From the graph, we can see that f(b) = 3.

Therefore, the value of the definite integral ∫︎[a, b] f'(x) dx is equal to 3.