Applications of derivatives · Definition

Mean value theorem

calculusapplicationstheorems

Before you start you need to know

This theorem says that between two points of a curve there is at least one point where the tangent is parallel to the secant. You will use it to deduce global properties from the derivative.

Definition

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there is a \(c\in(a,b)\) such that the slope of the tangent at \(c\) equals the slope of the secant line joining \((a,f(a))\) and \((b,f(b))\).

\[f'(c)=\frac{f(b)-f(a)}{b-a}\]

Both hypotheses are necessary: if \(f\) has a corner inside \((a,b)\) or a jump at \(a\) or \(b\), the value c may not exist. The theorem only guarantees that c exists, without saying where it is.

With \(f(x)=x^{2}\) on \([1,3]\), the secant has slope \(\frac{9-1}{3-1}=4\). Since \(f'(c)=2c\), we solve \(2c=4\) and get \(c=2\), which lies inside \((1,3)\).

Typical error: applying the theorem without checking the hypotheses. With \(f(x)=|x|\) on \([-1,1]\) the secant has slope \(0\), but \(f'(x)\) never equals \(0\) because \(f\) is not differentiable at \(0\).

Think about it

If f’(x)=0 for all x in an interval, what do you conclude?

That f is constant on that interval (by the mean value theorem).

Why does the mean value theorem generalize Rolle's theorem?

If \(f(a)=f(b)\), the chord slope is zero and the conclusion asks for \(f'(c)=0\), which is Rolle. Conversely, Rolle is the key step in the proof.

Exercises 20

  1. Find the value c guaranteed by the mean value theorem for \(f(x)=x^{2}\) on \([1,3]\).
  2. Find the value c guaranteed by the mean value theorem for \(f(x)=x^{3}\) on \([0,2]\).
  3. Find the value c guaranteed by the mean value theorem for \(f(x)=\sqrt{x}\) on \([1,4]\).
  4. Which hypothesis does the mean value theorem require for \(f\) on \([a,b]\)?
  5. Compute the slope of the secant line of \(f(x)=x^{3}\) between \(x=1\) and \(x=2\).
  6. According to the mean value theorem, what slope must the tangent at the point c have?
  7. Find the value c guaranteed by the mean value theorem for \(f(x)=x^{2}-4x\) on \([0,4]\).
  8. Find the value c guaranteed by the mean value theorem for \(f(x)=x^{3}-3x\) on \([0,2]\).
  9. If \(f'(x)=0\) for all \(x\) in \((a,b)\) and \(f\) is continuous on \([a,b]\), what can be said about \(f\)?
  10. Can the mean value theorem be applied to \(f(x)=|x|\) on \([-1,1]\)?
  11. For \(f(x)=x^2\) on \([1,3]\), which point \(c\) does the mean value theorem guarantee?
  12. For \(f(x)=x^3\) on \([0,2]\), which \(c\in(0,2)\) satisfies the conclusion of the theorem?
  13. For \(f(x)=\sqrt{x}\) on \([1,4]\), what is the point \(c\) of the mean value theorem?
  14. For \(f(x)=e^x\) on \([0,1]\), which \(c\) satisfies \(f'(c)=f(1)-f(0)\)?
  15. For \(f(x)=\ln x\) on \([1,e]\), what is the point \(c\) of the mean value theorem?
  16. Which hypotheses does the mean value theorem require?
  17. If \(f'(x)=0\) for every \(x\) in an interval \(I\), what follows?
  18. Can the mean value theorem be applied to \(f(x)=|x|\) on \([-1,2]\)?
  19. A differentiable function satisfies \(f'(x)\le3\) for all \(x\) and \(f(1)=2\). What is the best bound you can guarantee for \(f(5)\)?
  20. If \(g'(x)=2x\) on \(\mathbb R\) and \(g(1)=5\), what is \(g(3)\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §5.3, Teorema 5.3.2
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 5, Teorema 5.10
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §6.2, Teorema 6.2.4