MTH6131 Chap.8 Mean Value Theorem and Taylor Approximation
Mean Value Theorem and Taylor Approximation
Rolle's Theorem and the Mean Value Theorem
The notes state Rolle's Theorem first: a function that is unbroken on a closed interval, has a derivative at every interior point and vanishes at both ends must have a horizontal tangent somewhere inside.
The Mean Value Theorem drops the requirement of equal end values: under the same continuity and differentiability, the derivative somewhere inside equals the slope of the chord, (f(b) − f(a))/(b − a). Geometrically, some tangent is parallel to the chord.
The rearranged form and its two uses
Rearranged, the theorem says f(b) = f(a) + (b − a)f′(x₀) for some x₀ between a and b. The notes use this form in two ways.
First, to approximate: the exact x₀ is unknown, so it is frozen at a, which gives an estimate such as a sixth root of a number near a perfect sixth power. Second, to prove inequalities: bounding f′(x₀) on the interval bounds the change in f, which is how the notes prove that sin x is less than x for positive x.
Taylor's Theorem
Taylor's Theorem extends the idea with more derivatives.
When f and every derivative up to order n − 1 are unbroken on the interval, and the n-th derivative exists inside it, then f(b) equals a polynomial in (b − a) built from the derivatives at a, plus a remainder term involving the n-th derivative at some unknown point. With a = 0 this is the Maclaurin form.
The notes ask for an approximation of e by this route.
Error as evidence
What makes these theorems useful is that the error is controlled. A bound on the next derivative over the interval is a bound on the remainder, so an approximation comes with a guarantee of how far off it can be, and knowing whether the derivative is increasing or decreasing even tells you the direction of the error.
What this chapter covers
- 01
Rolle's Theorem and its hypotheses
- 02
The Mean Value Theorem as a parallel tangent
- 03
The rearranged form f(b) = f(a) + (b − a)f′(x₀)
- 04
Estimating roots by freezing the slope
- 05
Proving inequalities by bounding the derivative
- 06
Taylor's Theorem with remainder
- 07
Maclaurin polynomials
- 08
Bounding the approximation error
Worked example · free
A cube root from the Mean Value Theorem
- 2Write the theorem in the form f(28) = f(27) + f′(x₀).
- 2Freeze x₀ at 27 and evaluate.
- 1Use the monotonicity of f′ to decide the direction of the error.
Key terms
- Rolle's Theorem
- The result that a function continuous on a closed interval, differentiable inside it and equal at both ends has a zero derivative somewhere inside.
- Mean Value Theorem
- The result that a function continuous on a closed interval and differentiable inside it has, somewhere inside, a derivative equal to the slope of the chord.
- Maclaurin Polynomial
- A Taylor polynomial centred at zero, built from the derivatives of a function at the origin.
- Remainder Term
- The final term of Taylor's Theorem, which gives the exact error of the approximating polynomial at some unknown point.
Mean Value Theorem and Taylor Approximation FAQ
What does the Mean Value Theorem actually tell me?
It guarantees that somewhere between a and b the instantaneous rate of change equals the average rate over the interval, provided the function is continuous on the closed interval and differentiable inside it. It does not say where, which is why it is used for estimates and inequalities rather than for finding a specific point.
How do I know how accurate a Taylor approximation is?
Bound the remainder term. It involves the next derivative at an unknown point between the centre and the input, so find the largest value that derivative can take on that interval and substitute it. The result is a guaranteed upper limit on the error, and the true error will be no larger.
Why does the Mean Value Theorem fail for the absolute value on an interval containing zero?
Because the function is not differentiable at zero, so one hypothesis fails. On the interval from −1 to 1 the chord is horizontal, yet the slope is only ever −1 or 1, never zero. Checking both hypotheses before quoting the theorem avoids exactly this trap.
Exam move
For each approximation you make, write down the error bound and then compute the true error, and check that the true error is inside the bound. Doing this three or four times makes the remainder term feel like a tool rather than a formula to memorise.
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