Lagrange Error Bound

3. Visual Connection

5. Concept Development

BC-only topic. A Taylor polynomial gives a local approximation, but the Lagrange Error Bound turns that approximation into a guaranteed statement about accuracy.

Remainder notation

If \(P_n\) is the degree-\(n\) Taylor polynomial for \(f\) centered at \(a\), define

The quantity \(R_n(x)\) is the signed error, while \(|R_n(x)|\) is the absolute error.

Taylor's Theorem with Lagrange remainder

If \(f\) has the required derivatives on the interval between \(a\) and \(x\), then some number \(c\) between \(a\) and \(x\) satisfies

The unknown location \(c\) prevents direct calculation, so an interval-wide derivative bound is used.

Lagrange Error Bound

If

for every \(t\) between \(a\) and \(x\), then

Read every index carefully

• A degree-\(n\) polynomial requires a bound on the \((n+1)\)st derivative.
• The exponent on the distance is \(n+1\).
• The factorial is \((n+1)!\).

These three appearances of \(n+1\) must agree.

The relevant interval

The derivative bound must hold on the entire closed interval joining the center and target:

If \(x<a\), do not reverse the endpoints or use a negative distance; the formula contains \(|x-a|\).

How to choose M

Any valid upper bound for \(|f^{(n+1)}|\) on \(I\) works. A practical choice is the smallest easy bound you can justify.

• For sine and cosine derivatives, \(M=1\) is always valid.
• For \(e^t\), use the exponential value at the right endpoint because \(e^t\) increases.
• For a decreasing positive expression such as \(C/t^p\), inspect the left endpoint.
• If the problem provides \(|f^{(n+1)}(t)|\le K\), use \(M=K\).
• For a more complicated derivative, analyze critical points and endpoints of \(|f^{(n+1)}|\).

M is not usually a value at the center

The number \(|f^{(n+1)}(a)|\) is valid only if it also bounds the derivative everywhere between \(a\) and \(x\). Checking one point without a monotonicity or maximum argument does not justify \(M\).

A step-by-step workflow

1. Identify the polynomial degree \(n\), center \(a\), and target \(x\).
2. Find or identify \(f^{(n+1)}\).
3. Write the interval between \(a\) and \(x\).
4. Justify a bound \(M\) for \(|f^{(n+1)}|\) on that interval.
5. Substitute \(M\), \(|x-a|\), and \((n+1)!\) into the formula.
6. Compare the result with the requested tolerance.

The bound is conservative

The Lagrange expression is a guaranteed maximum, not the actual error. A convenient \(M\) may be larger than the true maximum, making the bound loose but still valid.

Uniform accuracy on an interval

If \(|f^{(n+1)}(t)|\le M\) and \(|x-a|\le d\) throughout a target interval, then one calculation gives

for every \(x\) in that interval.

Choosing a polynomial degree

To guarantee error at most \(\varepsilon\), find the least nonnegative integer \(n\) for which

where \(M_n\) bounds the derivative of order \(n+1\). Check successive degrees because \(M_n\) may change with \(n\).

What the bound does not determine

The absolute-value inequality alone does not tell whether \(P_n(x)\) is an overestimate or underestimate. That requires additional sign information about \(f^{(n+1)}(c)(x-a)^{n+1}\).

Lagrange versus alternating-series error

The Lagrange bound uses a higher-derivative maximum and applies to Taylor-polynomial remainders under Taylor's theorem. The alternating-series bound uses the first omitted term and requires alternating, decreasing magnitudes. Use the theorem whose hypotheses are established.

AP-style checklist

1. Name the degree and therefore the required derivative order.
2. State the interval between the center and target.
3. Explain why the selected \(M\) bounds the derivative on that whole interval.
4. Display the complete Lagrange inequality.
5. Keep the error in absolute value.
6. Finish by comparing the bound with the requested accuracy.

6. Detailed Worked Example and Error Check

Example 1: Exponential approximation to the right

Approximate \(e^{0.2}\) with the degree-\(3\) Maclaurin polynomial. Since \(f^{(4)}(t)=e^t\) and \(e^t\le e^{0.2}\) on \([0,0.2]\), take \(M=e^{0.2}\). Then

Example 2: Exponential approximation to the left

For the degree-\(2\) Maclaurin approximation of \(e^{-0.3}\), the interval is \([-0.3,0]\). Because \(e^t\le1\) there,

Example 3: Sine with a standard bound

The degree-\(3\) Maclaurin polynomial for \(\sin x\) is \(x-x^3/6\). Since \(f^{(4)}(t)=\sin t\) and \(|\sin t|\le1\),

Example 4: Cosine approximation

For the degree-\(4\) Maclaurin polynomial \(1-x^2/2+x^4/24\), the fifth derivative of cosine has magnitude at most \(1\). Therefore

Example 5: Logarithm centered at 1

Approximate \(\ln(1.1)\) with \(P_2\) centered at \(1\). Since \(f'''(t)=2/t^3\), its largest magnitude on \([1,1.1]\) is \(M=2\). Thus

Example 6: Square root with an endpoint maximum

For \(f(x)=\sqrt x\), \(f'''(t)=3/(8t^{5/2})\). On \([4,4.1]\), this is largest at \(t=4\), so \(M=3/256\). The degree-\(2\) approximation centered at \(4\) satisfies

Example 7: A derivative bound supplied by the problem

Suppose \(|f^{(5)}(t)|\le15\) for \(0\le t\le0.5\). Using the degree-\(4\) Maclaurin polynomial at \(x=0.1\),

This proves the requested accuracy without knowing the exact function value.

Example 8: One bound for an entire interval

For the degree-\(3\) Maclaurin polynomial of \(e^x\) on \([-0.5,0.5]\), use \(M=e^{0.5}\) and \(|x|\le0.5\):

Example 9: Find the least degree

To approximate \(e^{0.1}\) with error at most \(10^{-8}\), use \(M=e^{0.1}\) on \([0,0.1]\). Degree \(4\) gives

which is too large. Degree \(5\) gives \(e^{0.1}(0.1)^6/6!\approx1.54\times10^{-9}\), so the least guaranteed degree is \(5\).

Example 10: Why checking only the center fails

Let \(f(x)=1/(1-x)\), centered at \(0\), and use \(P_2\) at \(x=0.5\). Here \(f'''(t)=6/(1-t)^4\). Its center value is \(6\), but on \([0,0.5]\) the maximum is \(96\). Using \(M=6\) would produce an invalid bound \(0.125\), smaller than the actual error \(2-1.75=0.25\). The interval-wide bound \(M=96\) is valid, though conservative.

Common errors

• Bounding \(f^{(n)}\) instead of \(f^{(n+1)}\).
• Using \(n!\) or \(|x-a|^n\) in a degree-\(n\) error formula.
• Choosing \(M=|f^{(n+1)}(a)|\) without an interval argument.
• Evaluating the derivative only at the target point.
• Forgetting the absolute value on the derivative or error.
• Using \(x\) instead of the distance \(|x-a|\).
• Writing the interval in the wrong order when \(x<a\).
• Calling the upper bound the exact error.
• Assuming a larger \(M\) makes the argument invalid; it only makes the estimate less sharp.
• Claiming overestimate or underestimate from an absolute bound alone.
• Using the alternating-series error formula without verifying alternation and decrease.
• Rounding the bound before comparing it with the required tolerance.