AP® Calculus BC · Unit 10 · Topic 10.11
Maclaurin Polynomials and How Degree Improves the Fit
A Maclaurin polynomial approximates a function near x = 0 using its derivatives at that point. Adding higher-degree terms makes the polynomial match the function's shape over a wider interval, not just at the single point.
A Maclaurin polynomial is built so that it shares the function's value and as many derivatives as possible at x = 0. The degree 1 polynomial matches the function's value and slope at 0, giving a tangent line. The degree 3 polynomial also matches the second and third derivatives, so it starts bending the right way near 0. Each new term corrects an error the previous polynomial still had, and that correction grows in strength as x moves away from 0. That is why more terms let the fit hug the curve over a wider stretch: each added term handles the next layer of curvature the lower-degree version ignored.
Look at sin(x). The degree 1 polynomial is just x, a straight line that only matches sin(x) right at the origin. Degree 3 gives x - x3/6, which bends downward and tracks the sine wave through its first hump. Degree 5 and degree 7 add x5/120 and -x7/5040, each pulling the polynomial back toward the true curve a little longer before it veers off. The pattern: odd-degree terms only, alternating signs, factorial denominators, because sine's even derivatives at 0 are all zero.
This degree-chasing idea is the bridge to full Maclaurin series. A series is just this process carried to infinity, so the polynomial becomes an exact match everywhere the series converges. Later in the unit, series built this way get used to evaluate limits that resist L'Hopital's rule and to integrate functions with no elementary antiderivative, by swapping the function for its series and working term by term.
Key terms
- Maclaurin polynomial
- A polynomial that matches a function's value and derivatives at x = 0 up to some degree.
- Degree
- The highest power of x appearing in the polynomial, which sets how many derivatives are matched.
- Taylor coefficient
- The number fn(0)/n! multiplying xn in the polynomial, built from the nth derivative at 0.
- Maclaurin series
- The infinite extension of a Maclaurin polynomial, taking the degree to infinity.
Watch it explained
- How it worksHow do the degree 1, 3, 5, and 7 Maclaurin polynomials of sin(x) close in on the sine wave?
- Worked exampleFind the fourth-degree Maclaurin polynomial for f(x) = e2x.
- Worked exampleFind the degree 4 Maclaurin polynomial for cos(x) and evaluate it at x = 0.5.
- Common mistakeWhy doesn't a higher degree always mean a better approximation everywhere?
- Ties togetherHow does raising the degree connect to using a Maclaurin series to crack limits and integrals later in the unit?
Check yourself
End of topic.