AP Calculus AB Exam Question:

The rate of change of a function f(x) is given by f'(x) = 3x^2 - 8x + 5.

1. Determine the net change of f(x) over the interval [1, 4].

2. Find the accumulated change of f(x) from x = 1 to x = 4.

Solution:

1. To find the net change of a function over an interval, we integrate its derivative over that interval.

Integrating f'(x) = 3x^2 - 8x + 5 with respect to x, we get:

f(x) = ∫ (3x^2 - 8x + 5) dx

= x^3 - 4x^2 + 5x + C

Now, to find the net change over the interval [1, 4], we evaluate f(4) - f(1):

f(4) - f(1) = (4)^3 - 4(4)^2 + 5(4) + C - [(1)^3 - 4(1)^2 + 5(1) + C]

= 64 - 64 + 20 + C - (1 - 4 + 5 + C)

= 20 + C - 0

= 20

Therefore, the net change of f(x) over the interval [1, 4] is 20.

2. The accumulated change of f(x) from x = 1 to x = 4 is given by the definite integral of f'(x) over that range:

Accumulated change = ∫(f'(x)) dx from 1 to 4

= ∫(3x^2 - 8x + 5) dx from 1 to 4

= [ x^3 - 4x^2 + 5x ] from 1 to 4

= (4)^3 - 4(4)^2 + 5(4) - [(1)^3 - 4(1)^2 + 5(1)]

= 64 - 64 + 20 - (1 - 4 + 5)

= 20

Therefore, the accumulated change of f(x) from x = 1 to x = 4 is also 20.