Taylor Polynomials

A function is picked from the list or typed as in R. The green diamond is the center $a$. Press + or − to change the degree $n$. The worked column writes the Taylor polynomial in the general form:

$$T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n.$$

The lower degrees are drawn dashed. The gap between two consecutive curves is the term added at the higher degree. The magenta dot is the input $b$. The magenta bar at $b$ is the error $|f(b) - T_n(b)|$.

Show numbers gives $T_n$ with the derivative values substituted and a table of the matching derivatives at the center. Show numbers also gives the estimate $T_n(b)$, the true value $f(b)$, and the error. A last table lists the error at $b$ for each degree.

Under More, Lower degrees turns the dashed curves off and on. Newest term alone draws the newest term $\frac{f^{(n)}(a)}{n!}(x - a)^n$ as a dotted curve.

Show R gives the R code that builds $T_n$ with D() and factorial(). The code plots $T_n$ over $f$ and prints the estimate and the error.

Keys: press A or B to pick the center or the input. Press [ or ] to step through the handles. Press the left or right arrow key to move it one step. Hold Shift with an arrow key to move it five steps. Press Esc to let go.