Taylor’s Remainder
Error bound of Taylor approximations.
Key idea
Error bound of Taylor approximations.
In depth
Rₙ(x) = f⁽ⁿ⁺¹⁾(c)(x − a)ⁿ⁺¹/(n + 1)! for some c between a and x (Lagrange form). It bounds the error of an nth-degree Taylor polynomial and proves that series converge to the function.
Worked example
Approximating e⁰·¹ with 3 terms: error ≤ e⁰·¹(0.1)³/6 < 0.0002.
Real-world uses
- Stating how accurate a Taylor approximation is in numerical software and engineering safety margins
Common mistake
Treating the polynomial as exact.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Taylor’s Remainder?
Error bound of Taylor approximations. It is a topic at the expert level in Calculus.
Where is Taylor’s Remainder used in real life?
Real-world examples: Stating how accurate a Taylor approximation is in numerical software and engineering safety margins.
What should I learn before Taylor’s Remainder?
Make sure you are comfortable with Common Maclaurin Series, the previous topic in Calculus.
What is a common mistake with Taylor’s Remainder?
Treating the polynomial as exact.
How can I practise Taylor’s Remainder?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Taylor’s Remainder?
Move on to Parametric Derivatives next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.