AP Calculus AB Exam Question

Let f(x) be a continuous function defined on the closed interval [1, 5]. The function f(x) is shown in the graph below:

(a) Use geometry to estimate the average value of f(x) over the interval [1, 5]. Show your work.

(b) Verify your estimate for the average value of f(x) in part (a) by evaluating the definite integral of f(x) over [1, 5]. Show all steps of your work.

Answer:

(a) To estimate the average value of f(x) over the interval [1, 5] using geometry, we will draw rectangles to approximate the area under the curve.

First, divide the interval [1, 5] into n equal subintervals. We will use n = 4 in this case for simplicity.

The width of each rectangle will be Δx = (5-1)/4 = 1. The height of each rectangle will be the midpoint value of f(x) within each subinterval.

Subinterval 1: [1, 2] Midpoint of f(x) in this subinterval is (1+2)/2 = 1.5. The rectangle has width 1 and height f(1.5).

Subinterval 2: [2, 3] Midpoint of f(x) in this subinterval is (2+3)/2 = 2.5. The rectangle has width 1 and height f(2.5).

Subinterval 3: [3, 4] Midpoint of f(x) in this subinterval is (3+4)/2 = 3.5. The rectangle has width 1 and height f(3.5).

Subinterval 4: [4, 5] Midpoint of f(x) in this subinterval is (4+5)/2 = 4.5. The rectangle has width 1 and height f(4.5).

Now, find the areas of each rectangle and sum them up:

Estimated average value of f(x) = (1 * f(1.5) + 1 * f(2.5) + 1 * f(3.5) + 1 * f(4.5))/4

(b) To verify our estimate for the average value of f(x) in part (a), we need to evaluate the definite integral of f(x) over [1, 5].

∫[1, 5] f(x) dx = F(5) - F(1), where F(x) is the antiderivative of f(x).

Now, we'll find F(x):

∫[1, 5] f(x) dx = [F(x)] evaluated from 1 to 5 = F(5) - F(1)

Therefore, the average value of f(x) over the interval [1, 5] is equal to:

Average value of f(x) = (F(5) - F(1))/(5-1)

Evaluate the definite integral and calculate the average value using the fundamental theorem of calculus.

Note: Provide the necessary information such as the graph of f(x) or the equation of f(x) to proceed with the solution.