AP® Calculus BC · Unit 10 · Topic 10.11
Finding Taylor Polynomial Approximations of Functions
A Taylor polynomial is built by matching a function's value and derivatives at one point, so it mimics the function's shape near that point. The more derivatives you match, the more terms you add and the better the fit.
A polynomial is completely determined once you know its value and all its derivatives at a single point. Taylor polynomials exploit this: pick a center x = c, and build a polynomial whose value, slope, concavity, and so on all agree with f at c. Each coefficient is fn(c) / n!, so the polynomial is T(x) = f(c) + f'(c)(x-c) + f''(c)/2! (x-c)2 + f'''(c)/3! (x-c)3 + ... Matching more derivatives means adding one more term, and each new term is built to fix exactly one more derivative at c without disturbing the ones already matched.
Think of it as zooming in at x = c. The degree-0 term matches height. The degree-1 term matches slope, giving the tangent line. The degree-2 term matches concavity, curving the graph correctly. Every extra term refines the shape a little more, but only the information available at c. That is why the approximation is sharp near c and gets worse as x moves away: the polynomial knows nothing about how f actually behaves far from its center, only how it behaves right at that point.
This sets up the next idea directly: since the polynomial is only as good as the derivatives it captures, there must be a way to measure what's left out. That leftover is the remainder, and bounding it with the Lagrange error bound tells you how large the gap between f and its Taylor polynomial can be. Higher degree means more matched derivatives, which generally means a tighter, smaller guaranteed error, especially close to the center.
Key terms
- Taylor polynomial
- A polynomial built from a function's derivatives at one point, used to approximate the function near that point.
- Center
- The point x = c where the Taylor polynomial's derivatives are matched to the function's derivatives.
- Degree
- The highest power of (x - c) in the polynomial, equal to the number of derivatives matched.
- Remainder
- The difference between the actual function and its Taylor polynomial at a given x.
- Lagrange error bound
- A formula that bounds how large the remainder can be, based on a bound on the next derivative.
Watch it explained
- How it worksHow does each new Taylor term fix one more derivative at the center?
- Worked exampleFind the third-degree Taylor polynomial for f(x) = ln(x) centered at x = 1.
- Worked exampleFind the second-degree Taylor polynomial for f(x) = √x centered at x = 4.
- Common mistakeWhy isn't a Taylor polynomial a good approximation far away from its center?
- Ties togetherHow does increasing the degree of this Taylor polynomial connect to how tightly the Lagrange error bound pins down its leftover error?
Check yourself
End of topic.