MATHEMATICS LAB
Taylor and Maclaurin Series
Build a Taylor polynomial, compare degrees, and visualize the approximation error.
Polynomial and approximation
Change the degree to build the polynomial one term at a time. The error may increase: a higher degree does not always give a better approximation.
Building the term
Show Pₖ₋₁(x) and the new polynomial
Show the symbolic derivative of this order
Graphical comparison
Extended view: compare the function and polynomial away from the expansion center.
Auto scaling includes all finite samples from both curves. Vertical zoom may clip some of them; any clipping is reported below the graph.
Absolute error across the interval
The graph shows |f(t) − Pₙ(t)| over the same horizontal range, with automatic or adjustable vertical scaling. If zooming clips samples, their number is reported. Points outside the function’s domain are excluded.
Accuracy and error
Actual error at the selected point
Theoretical remainder
Error on the interval from center to point
The actual error refers to one point; a uniform upper bound applies throughout the stated interval. The largest sampled error in the graph does not prove the true maximum on that interval.
Convergence of the infinite series
At the selected point x
Behavior at the endpoints
A finite-degree Taylor polynomial and an infinite Taylor series are not the same thing.
Coefficient table
One term per row. Select a degree to examine the partial polynomial and how it is constructed.
| Degree k | Coefficient cₖ = f⁽ᵏ⁾(a)/k! | Added term | Pₖ(x) at the selected point | Absolute error |
|---|
The † symbol in the error column marks a value rounded to zero, not an exactly zero mathematical remainder. Coefficients are computed using truncated power-series algebra. An explicit derivative is displayed only when it can be obtained in readable form; a generic placeholder is never presented as an explicit derivative.
How the method works
- Choose the expansion center a and maximum degree n.
- Compute the coefficients cₖ = f⁽ᵏ⁾(a)/k! using truncated series, equivalent to the derivative formula.
- Add the term cₖ(x − a)ᵏ to the previous polynomial.
- Compare the values at the chosen point, distinguishing numerical error, theoretical remainder bounds, and convergence of the infinite series.
If the center is a = 0, the expansion is a Maclaurin series. A finite Taylor polynomial does not guarantee that the infinite series converges or equals the function away from the center. At nonsmooth points (for example, abs(x) and sqrt(x) at zero), the calculator does not fabricate unjustified expansions.