Mean Value Theorem Math Example 1

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

Example 1

easy
Verify the MVT for f(x)=x2f(x) = x^2 on [1,3][1, 3] by finding the value of cc.

Solution

  1. 1
    Check hypotheses: ff is a polynomial, so continuous on [1,3][1,3] and differentiable on (1,3)(1,3). ✓
  2. 2
    Average rate of change: f(3)−f(1)3−1=9−12=4\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4.
  3. 3
    Find cc: f′(x)=2xf'(x) = 2x. Set 2c=4⇒c=22c = 4 \Rightarrow c = 2.
  4. 4
    Check: c=2∈(1,3)c = 2 \in (1, 3). ✓

Answer

c=2∈(1,3)c = 2 \in (1, 3), confirming the MVT.
The MVT guarantees a cc where the instantaneous rate equals the average rate. Here c=2c=2 is where the tangent to y=x2y = x^2 is parallel to the secant through (1,1)(1,1) and (3,9)(3,9).

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 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].

Example 4 medium

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

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