From a Derivative Table to a Polynomial
A derivative table can make a Taylor-polynomial problem look like a matter of copying values into a formula. The essential steps are more precise: identify the center, use only the derivatives required by the degree, divide each derivative by the appropriate factorial, and attach the corresponding power of the distance from the center.
The center matters because the polynomial is built from the function’s behavior at one particular input \(a\). Its terms are powers of \(x-a\), not generally powers of \(x\). When \(x\) is near \(a\), the shift \(x-a\) is small, which is why this polynomial can serve as a local model.
A derivative table for a center \(a\) lists the numbers \(f(a), f'(a), f''(a),\ldots\). To form a degree-\(n\) polynomial, use the entries from order \(0\) through order \(n\). The entry in row \(k\) becomes the coefficient of \((x-a)^k\) only after division by \(k!\). An entry beyond order \(n\) does not contribute to the requested polynomial.
Why the Formula Uses Factorials
The factorials are required because differentiating a power \(k\) times introduces a factor of \(k!\). For instance, the \(k\)th derivative of \((x-a)^k\) is the constant \(k!\). The coefficient \(f^{(k)}(a)/k!\) therefore makes the \(k\)th derivative of that term equal to \(f^{(k)}(a)\) at the center.
More generally, the formula is precisely the one that makes the polynomial’s derivatives at \(a\) agree with the available table entries. To see why, write a polynomial in powers of \(x-a\) as \(P(x)=\sum_{j=0}^{n}c_j(x-a)^j\). At \(x=a\), every positive power vanishes, so \(P(a)=c_0\). For a derivative of order \(k\), terms with \(j<k\) have already differentiated to zero, and terms with \(j>k\) still contain a positive power of \(x-a\) and vanish at \(a\). The \(j=k\) term contributes \(k!c_k\). Thus \(P^{(k)}(a)=k!c_k\), and matching the table requires \(c_k=f^{(k)}(a)/k!\).
Worked Examples
Worked Example: A Cubic Centered at 2
A function has the following derivative values at \(a=2\). Find its degree-3 Taylor polynomial and use it to estimate the function at \(x=2.1\).
| Derivative order \(k\) | \(f^{(k)}(2)\) |
|---|---|
| 0 | 1 |
| 1 | \(-2\) |
| 2 | 6 |
| 3 | \(-24\) |
For degree 3, use the entries through order 3. The corresponding terms are
At \(x=2.1\), the shift from the center is \(h=x-2=0.1\). Substituting that shift gives
Check the polynomial against the table by differentiating with respect to \(x\). Its first three derivatives are \(T_3'(x)=-2+6(x-2)-12(x-2)^2\), \(T_3''(x)=6-24(x-2)\), and \(T_3'''(x)=-24\). At \(x=2\), these give \(-2\), \(6\), and \(-24\), while \(T_3(2)=1\). All four values match the table.
Worked Example: A Fourth-Degree Polynomial Centered at \(-1\)
Suppose the table at \(a=-1\) gives \(f(-1)=2\), \(f'(-1)=3\), \(f''(-1)=-4\), \(f'''(-1)=0\), and \(f^{(4)}(-1)=24\). Construct the degree-4 Taylor polynomial and evaluate it at \(x=-0.8\).
Write each term using \(x-a=x-(-1)=x+1\), and divide each derivative value by its factorial:
The input \(x=-0.8\) is \(0.2\) units to the right of the center, so \(x+1=0.2=1/5\). Then
For a check, set \(h=x+1\). Then \(T_4=2+3h-2h^2+h^4\). Differentiation gives \(T_4'=3-4h+4h^3\), \(T_4''=-4+12h^2\), \(T_4'''=24h\), and \(T_4^{(4)}=24\). At the center \(x=-1\), we have \(h=0\), so the polynomial and its first four derivatives give \(2,3,-4,0,24\), respectively, as required.
Worked Example: Use the Requested Degree, Not Every Table Entry
At \(a=3\), a derivative table gives \(f(3)=5\), \(f'(3)=-1\), \(f''(3)=8\), and \(f'''(3)=30\). Find the degree-2 Taylor polynomial and evaluate it at \(x=3.05\).
The requested degree is 2, so use only the entries through \(f''(3)\). The third derivative is not part of this polynomial:
Here \(x-3=3.05-3=0.05=1/20\). Therefore,
To check, \(T_2'(x)=-1+8(x-3)\) and \(T_2''(x)=8\). At \(x=3\), \(T_2(3)=5\), \(T_2'(3)=-1\), and \(T_2''(3)=8\), matching the entries used. Its third derivative is zero, not 30. That does not invalidate the degree-2 polynomial: it is not intended to match the function’s third derivative.
A Reliable Process for Table Problems
Writing the shift \(h=x-a\) can make the construction and evaluation easier to organize. It is only a temporary name for the distance from the center; the resulting polynomial should still be written in powers of \(x-a\).
Record \(a\) and \(n\). These determine the shift \(x-a\) and the highest derivative to use.
Use \(f(a)\) and \(f^{(k)}(a)\) for \(1\leq k\leq n\). Do not include higher-order entries in a degree-\(n\) polynomial.
Divide the order-\(k\) entry by \(k!\), then multiply by \((x-a)^k\).
For a target input \(x\), calculate \(h=x-a\) and substitute \(h\) into the polynomial. Keep exact fractions when convenient.
Differentiate the polynomial as needed and evaluate at \(x=a\). Its derivatives through the requested degree must reproduce the table entries used.
A Common Misread of the Table
A frequent error is to use a derivative value directly as a coefficient. In the first example, the table entry \(f''(2)=6\) is not the coefficient of \((x-2)^2\). The incorrect term would be \(6(x-2)^2\). It would contribute \(12\), rather than \(6\), to the polynomial’s second derivative at \(x=2\), because the second derivative of \((x-2)^2\) is 2.
The correction is to divide by \(2!\): the correct term is \(\frac{6}{2!}(x-2)^2=3(x-2)^2\), whose second derivative at \(x=2\) is \(3\cdot2=6\). In general, skipping the factorial makes the polynomial’s derivatives at the center fail to match the table.
Using the Polynomial as a Local Model
Once constructed, a Taylor polynomial gives a number that can be used as an estimate of \(f(x)\) near the center. The examples show the computation of that polynomial value; they do not, by themselves, establish the size of the estimation error. A derivative table through degree \(n\) determines the degree-\(n\) polynomial, but it does not provide a numerical error bound for a particular input.
For table problems, keep the tasks distinct: the requested degree specifies which derivatives to use; the center specifies the powers in the polynomial; and the target input determines the shift used for evaluation. Checking the polynomial’s derivatives at the center confirms that the construction matches the table data required for that degree.
Check Your Understanding
Use the Taylor-polynomial formula and check your work at the center.
- At \(a=1\), suppose \(f(1)=2\), \(f'(1)=4\), and \(f''(1)=6\). Write the degree-2 Taylor polynomial in powers of \(x-1\).
- If \(f^{(3)}(a)=18\), what is the coefficient of \((x-a)^3\) in the Taylor polynomial?
- A table gives derivatives through order 4, but the problem requests degree 2. Which entries are used, and why?
- For a polynomial centered at \(a=-2\), what shift should be substituted when evaluating at \(x=-1.9\)?
- Why does checking the polynomial’s derivatives at the center test whether its coefficients were constructed correctly?