Mean Value Theorem Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
A car travels 120 miles in 2 hours. Explain why the MVT guarantees the car exceeded 60 mph at some instant.

Solution

  1. 1
    Let s(t)s(t) be position (miles) at time tt (hours). ss is continuous and differentiable.
  2. 2
    Average speed: s(2)−s(0)2−0=1202=60\frac{s(2)-s(0)}{2-0} = \frac{120}{2} = 60 mph.
  3. 3
    By MVT, ∃ c∈(0,2)\exists\, c \in (0,2) such that s′(c)=60s'(c) = 60 mph.
  4. 4
    So the speedometer read exactly 60 mph at least once.

Answer

By MVT, the car's instantaneous speed equaled 60 mph at some time c∈(0,2)c \in (0, 2).
The MVT is essentially the mathematical version of the statement: 'If your average speed is 60 mph, your speedometer must have read exactly 60 at some moment.'

About Mean Value Theorem

If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point cc in (a,b)(a, b) where f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

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More Mean Value Theorem Examples

Example 1 easy

Verify the MVT for [formula] on [formula] by finding the value of [formula].

Example 2 hard

Use the MVT to prove: if [formula] for all [formula] in [formula], then [formula] is constant on [fo

Example 3 easy

Find the value of [formula] guaranteed by the MVT for [formula] on [formula].

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