- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
- Theory
- CLP-1 §2.13Key results
The Mean Value Theorem practice question 3
A car travels 150 km along a highway in exactly 1.5 hours. Its position is a differentiable function of time. According to the Mean Value Theorem, there must be some instant at which the car’s speedometer reads exactly what value?
Give the answer in kilometres per hour.
The theorem says that at some instant the instantaneous rate equals the average rate over the whole trip.
Average speed is total distance over total time: divide 150 km by 1.5 h.
A number.
Theory CLP-1 §2.13Key results
Worked solution
Strategy
The average velocity over the trip is total distance over total time. The MVT says the instantaneous velocity equals that average at least once.
Working
Average velocity.
Position s(t) is differentiable, so by the MVT there is a time c with s′(c) = 100: the speedometer reads exactly 100 km/h at that instant.
Answer
The answer is 100 kilometres per hour.
Check
100 km/h held for 1.5 h covers exactly the 150 km travelled.
Where this goes wrong
Reading the theorem as “the car never exceeded 100”. It says the speed was exactly 100 at some moment; the car may well have been faster and slower at others.
Leans on
Average velocity, from The definition of the derivative
Saved in this browser only.