f_avg = (1/3) ∫ from 0 to 3 of x² dx = (1/3)(27/3) = (1/3)(9) = 3. The value 4.5 is the average of the endpoint values 0 and 9, and 9 is the integral before dividing by the length 3.
Question 2 Short Answer
For f(x) = x² on [0, 3], find a number c in [0, 3] with f(c) equal to the average value.
Think about your answer, then reveal below.
Model answer: c = √3 ≈ 1.732
The average value is 3, so solve c² = 3 to get c = √3, which lies in [0, 3]. The Mean Value Theorem for Integrals guarantees such a c exists because x² is continuous.
Question 3 True / False
The average value of f on [a, b] equals (f(a) + f(b))/2 for every continuous function f.
TTrue
FFalse
Answer: False
That formula holds for linear functions but not in general. For x² on [0, 3] it gives 4.5, while the integral formula gives 3. The integral accounts for every value of f on the interval, not just the endpoints.
Question 4 Multiple Choice
A car's velocity is v(t) = 6t meters per second for 0 ≤ t ≤ 4. What is its average velocity over this time?
A12 m/s
B24 m/s
C48 m/s
D6 m/s
Average velocity = (1/4) ∫ from 0 to 4 of 6t dt = (1/4)(3t² from 0 to 4) = (1/4)(48) = 12 m/s. The integral 48 is the total distance in meters, and dividing by 4 seconds gives the average.