Topic 10.11 Finding Taylor Polynomial Approximations of Functions: construct the Taylor (or Maclaurin) polynomial of a function about a center using its derivatives at that point (BC).

What this topic is asking

The College Board (Topic 10.11, BC only) builds the Taylor polynomial, a polynomial that matches a function's value and several derivatives at a chosen center, giving an excellent local approximation. It generalizes the tangent-line (linear) approximation of Unit 4 to higher degree, adding curvature, concavity, and beyond.

The formula

Why each coefficient is a derivative over a factorial

The structure $\frac{f^{(k)}(a)}{k!}$ is exactly what makes $P_n$ match $f$'s derivatives at the center. Differentiating $(x-a)^k$ exactly $k$ times gives $k!$, and the division by $k!$ cancels it so that $P_n^{(k)}(a) = f^{(k)}(a)$. In other words, the polynomial is engineered so its value, slope, concavity, and higher derivatives at $a$ agree with the function's. This is the higher-order generalization of the tangent line: the degree-1 polynomial $f(a) + f'(a)(x - a)$ is precisely the linearization from Unit 4, matching value and slope; degree 2 adds the $\frac{f''(a)}{2}(x-a)^2$ term to match concavity; and so on. Each extra degree captures one more derivative, tightening the fit near $a$.

A worked Maclaurin polynomial

Building from a table of derivatives

A frequent free-response format gives you the derivative values $f(a), f'(a), f''(a), \ldots$ at the center, often in a table, rather than a formula for $f$. You then plug these directly into $\frac{f^{(k)}(a)}{k!}(x - a)^k$ without computing any derivatives yourself, as in the worked exam question. The watchpoints are the factorials ($2! = 2$, $3! = 6$, $4! = 24$) and the powers of $(x - a)$ about the correct center, not about $0$ unless $a = 0$. Writing the general term first, then substituting each given value, keeps the bookkeeping straight and avoids dropping a factorial.

Using the polynomial to approximate values and derivatives

Once you have $P_n(x)$, you approximate $f$ at a nearby point by evaluating $P_n$ there, as in estimating $\cos(0.2)$ or $f(2.1)$ above. The approximation is best when $x$ is close to the center $a$, and accuracy improves with higher degree. You can also read derivative information off the polynomial: the coefficient of $(x-a)^k$ equals $\frac{f^{(k)}(a)}{k!}$, so multiplying by $k!$ recovers $f^{(k)}(a)$. Conversely, given a Taylor polynomial you can find, say, $f''(a)$ by taking the $(x-a)^2$ coefficient and multiplying by $2!$. This two-way relationship between coefficients and derivatives is a common exam twist.