Lagrange Error Bound Calculator
Enter a function, degree, center point, and evaluation point to bound the Taylor polynomial error.
Hi, welcome to Hive Calculator! When you approximate a function with a Taylor polynomial, the Lagrange error bound tells you the maximum possible size of the mistake you're making. Our Lagrange Error Bound Calculator finds that bound instantly for common functions, or for any function if you already know the maximum derivative value.
What Is the Lagrange Error Bound?
When a Taylor polynomial of degree n, centered at
a, is used to approximate f(x), the difference
between the true value and the approximation is
called the remainder, Rn(x). The Lagrange
error bound gives an upper limit on how large that
remainder can possibly be, without needing to know
its exact value.
This is extremely useful in calculus: it lets you
guarantee how accurate a Taylor polynomial
approximation is, even when you can't compute the
exact error directly.
The Lagrange Error Bound Formula
If M is the maximum value of |f⁽ⁿ⁺¹⁾(t)| for t
between the center a and the evaluation point x,
then:
|Rₙ(x)| ≤ M / (n + 1)! × |x − a|⁽ⁿ⁺¹⁾
Where n is the degree of the Taylor polynomial, a is
the center point, x is the point where you're
evaluating the approximation, and (n + 1)! is the
factorial of n + 1.
For a few common functions, M has a simple, known
form:
- sin(x) and cos(x): every derivative is a sine or cosine, so M = 1 always works.
- eˣ: derivatives are all eˣ, which is increasing, so M = e raised to the larger of a and x.
- ln(1 + x): the (n+1)th derivative shrinks as x grows, so M uses the smaller of a and x.
- 1 / (1 − x): the (n+1)th derivative grows as x approaches 1, so M uses the larger of a and x.
How to Use the Lagrange Error Bound Calculator
Using our Lagrange Error Bound Calculator is straightforward.
Choose a Function:
Select sin(x), cos(x), eˣ, ln(1 + x), or 1 / (1 − x), or pick
"Other" if your function isn't listed.
Enter the Degree (n), Center Point (a), and Evaluation Point (x):
These define the Taylor polynomial and where you're checking
its accuracy.
Check the Max of (n + 1)th Derivative, M:
For the five built-in functions, M is auto-filled for you as
soon as n, a, and x are entered. You can always type your own
value here to override it — this is required if you
picked "Other."
Click ‘Calculate’:
The calculator plugs M into the formula and computes the
Lagrange error bound instantly.
Clear or Share:
Use Clear to reset every field, or the Share icon to send your
result to others.
Example Calculation
Bound the error of the degree-4 Taylor polynomial for sin(x), centered at a = 0, evaluated at x = 1.
Since every derivative of sin(x) is a sine or cosine, M = 1.
|x − a| = |1 − 0| = 1
(n + 1)! = 5! = 120
|R₄(1)| ≤ 1 / 120 × 1⁵ = 0.008333
So the degree-4 Taylor polynomial for sin(x) at x = 1 is guaranteed accurate to within about 0.008333.
Common Mistakes to Avoid
When working with the Lagrange error bound manually, people often:
- Use n! instead of (n + 1)! in the denominator
- Forget that M must bound the (n + 1)th derivative, not the nth derivative
- Pick M from the wrong endpoint of the interval between a and x
Using the Hive Calculator eliminates all these issues and computes the bound precisely.
Why Use Hive Calculator's Lagrange Error Bound Calculator?
- Auto-filled M for sin(x), cos(x), eˣ, ln(1 + x), and 1 / (1 − x)
- M is always editable, so you can enter your own value for any function
- Instant, accurate results with clear, labeled steps
- Free to use — no signup or download required
It gives a guaranteed upper limit on the difference between a function and its Taylor polynomial approximation at a specific point. The true error could be smaller, but it will never be larger than this bound.
Choose "Other" and type your own value into the Max of (n + 1)th Derivative field — the maximum value of |f⁽ⁿ⁺¹⁾(t)| for t between a and x. This value is usually found by hand or given in your problem statement.
Every derivative of sin(x) and cos(x) is itself a sine or cosine function, and both are always between −1 and 1. So |f⁽ⁿ⁺¹⁾(t)| never exceeds 1 for any t or any derivative order, making M = 1 a valid bound in every case.