# Finding Taylor polynomial approximations of functions

> AP Calculus BC · Unit 10: Infinite Sequences and Series
> Source: https://www.owlsprep.com/study/ap-calculus-bc-u10-finding-taylor-polynomial-approximations-of/

This subtopic teaches you to build Taylor polynomials, which approximate smooth functions near a given center point. You will learn how to calculate these polynomials from derivatives, and recognize common examples that frequently appear on the AP BC exam.

**Prerequisites:** [Higher order derivatives](https://www.owlsprep.com/study/ap-calculus-bc-higher-order-derivatives/); [Power series basics](https://www.owlsprep.com/study/ap-calculus-bc-power-series-introduction/)

## Learning objectives

- State the definition of an nth-degree Taylor polynomial centered at a point $a$
- Calculate Taylor polynomials for any function by computing derivatives at the center
- Use known Maclaurin polynomials of common functions to construct new polynomials quickly

## Definition of a Taylor Polynomial

**nth-degree Taylor polynomial centered at $a$** — The unique nth-degree polynomial that matches the value of $f(x)$ and its first $n$ derivatives at $x=a$, given by the formula:

*Notation:* T_n(x)

*Example:* The first-degree Taylor polynomial of $f(x)$ at $a$ is exactly the tangent line approximation you learned earlier in calculus.

$$T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k$$

> **info**
>
> A Taylor polynomial centered at $a=0$ is called a **Maclaurin polynomial**. This is the most common type you will encounter on the AP exam.

**Worked example:** Find the 0th-degree Taylor polynomial of $f(x) = e^x$ centered at $a=1$.

1. For $n=0$, the sum only includes the $k=0$ term. Recall that the 0th derivative of $f$ is $f$ itself, and $0! = 1$.
2. Evaluate $f^{(0)}(a) = f(1) = e^1 = e$
3. Substitute into the Taylor formula:
4. $$T_0(x) = \frac{f(1)}{0!} (x-1)^0 = \frac{e}{1} \cdot 1 = e$$

## Step-by-Step Calculation of Taylor Polynomials

To avoid common mistakes, follow this consistent process to build any Taylor polynomial:

1. Compute all derivatives of $f(x)$ up to the $n$th derivative
2. Evaluate each derivative at the center point $x=a$
3. Divide each evaluated derivative by the corresponding factorial $k!$ for the $k$th term
4. Multiply each term by $(x-a)^k$ and add all terms together

**Worked example:** Find the 3rd-degree Taylor polynomial of $f(x) = \ln(x)$ centered at $a=1$.

1. Compute derivatives up to 3rd order:
2. $$f(x) = \ln x, \quad f'(x) = \frac{1}{x}, \quad f''(x) = -\frac{1}{x^2}, \quad f'''(x) = \frac{2}{x^3}$$
3. Evaluate each derivative at $a=1$:
4. $$f(1) = 0, \quad f'(1) = 1, \quad f''(1) = -1, \quad f'''(1) = 2$$
5. Calculate coefficients for each term from $k=0$ to $k=3$:
6. $$c_0 = \frac{0}{0!} = 0, \quad c_1 = \frac{1}{1!} = 1, \quad c_2 = \frac{-1}{2!} = -\frac{1}{2}, \quad c_3 = \frac{2}{3!} = \frac{1}{3}$$
7. Write the full polynomial:
8. $$T_3(x) = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3}$$

**Check your understanding**

Test your understanding of coefficient calculation:

1. What is the coefficient of $(x-2)^4$ in the 4th-degree Taylor polynomial of $f(x) = x^4$ centered at $a=2$?

   - 1
   - 4
   - 24
   - \frac{1}{24}

   *Why:* The 4th derivative of $f(x)=x^4$ is $4! = 24$. The coefficient is $\frac{f^{(4)}(2)}{4!} = \frac{24}{24} = 1$.

## Common Maclaurin Polynomials

Maclaurin polynomials (Taylor at $a=0$) for basic functions appear constantly on the AP exam. You can use these standard forms to quickly construct new polynomials for related functions by substitution, instead of recalculating derivatives from scratch.

| Function | General nth-degree Maclaurin Polynomial |
| --- | --- |
| $e^x$ | $1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!}$ |
| $\sin x$ | $x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... + (-1)^k \frac{x^{2k+1}}{(2k+1)!}$ |
| $\cos x$ | $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ... + (-1)^k \frac{x^{2k}}{(2k)!}$ |
| $\frac{1}{1-x}$ | $1 + x + x^2 + ... + x^n$ |
| $\ln(1+x)$ | $x - \frac{x^2}{2} + \frac{x^3}{3} - ... + (-1)^{n+1} \frac{x^n}{n}$ |

> **Alternating Sign Mnemonic**
>
> $\sin x$ alternates starting with positive for the first (x) term, $\cos x$ alternates starting with positive for the constant term, and $\ln(1+x)$ alternates starting with positive for the x term.

**Worked example:** Find the 4th-degree Maclaurin polynomial for $f(x) = e^{-2x}$ using the standard form for $e^u$.

1. Start with the 4th-degree Maclaurin for $e^u$:
2. $$e^u \approx 1 + u + \frac{u^2}{2!} + \frac{u^3}{3!} + \frac{u^4}{4!}$$
3. Substitute $u = -2x$ into the expression:
4. $$1 + (-2x) + \frac{(-2x)^2}{2} + \frac{(-2x)^3}{6} + \frac{(-2x)^4}{24}$$
5. Simplify each term to get the final polynomial:
6. $$T_4(x) = 1 - 2x + 2x^2 - \frac{4}{3}x^3 + \frac{2}{3}x^4$$

## Common pitfalls

- **Wrong:** Forgetting to divide the kth derivative by $k!$ when calculating the coefficient.
  - Why it fails: The Taylor formula requires division by factorial to match the derivative of the polynomial. Skipping this step gives a coefficient that is off by a factor of $k!$.
  - Correct: Always write the $k!$ denominator explicitly for each term, and double-check factorial values (e.g. $3! = 6$, not 3).
- **Wrong:** Writing $x^k$ instead of $(x-a)^k$ for a Taylor polynomial centered at $a \neq 0$.
  - Why it fails: Confusing Taylor and Maclaurin polynomials leads to an approximation that is only accurate near 0, not near the required center.
  - Correct: Always confirm the center first, and write $(x-a)$ for every non-constant term before simplifying.
- **Wrong:** Making sign errors when taking repeated derivatives of trigonometric or logarithmic functions.
  - Why it fails: Signs flip every time you take a derivative of $\sin x$, $\cos x$, or negative powers of $x$, so it is easy to end up with the wrong sign.
  - Correct: Write out each derivative one at a time, checking the sign and exponent at each step before evaluating at $a$.
- **Wrong:** Stopping at the $(n-1)$th term for an nth-degree polynomial, giving you only $n$ total terms.
  - Why it fails: Since counting starts at $k=0$, you need $n+1$ total terms to get an nth-degree polynomial.
  - Correct: Count your terms after constructing the polynomial to confirm you have $n+1$ terms.
- **Wrong:** When substituting into a known Maclaurin, forgetting to raise the entire substituted expression to the correct power.
  - Why it fails: For example, substituting $3x$ into $\sin x$ often leads to the incorrect term $\frac{3x^3}{3!}$ instead of $\frac{(3x)^3}{3!}$.
  - Correct: Always wrap the substituted expression in parentheses before raising it to the term power.

## Cheatsheet

| Concept | Key Formula | Exam Note |
| --- | --- | --- |
| nth-degree Taylor (center $a$) | $T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k$ | Matches $f$ and first $n$ derivatives at $a$ |
| nth-degree Maclaurin | $T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k$ | Taylor centered at $a=0$ |
| $e^x$ Maclaurin | $1 + x + \frac{x^2}{2!} + ... + \frac{x^n}{n!}$ | No alternating signs |
| $\sin x$ Maclaurin | $x - \frac{x^3}{3!} + \frac{x^5}{5!} - ...$ | Only odd powers, alternating signs |
| $\cos x$ Maclaurin | $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - ...$ | Only even powers, alternating signs |

## What's next

Taylor polynomials are the foundation for all power series approximations in AP Calculus BC. Once you master how to construct them, you can move on to studying the error of these approximations using Lagrange error bounds, and extend finite Taylor polynomials to infinite Taylor series that converge to the original function on an interval of convergence. Taylor polynomials also have wide applications beyond the exam, used in physics, engineering, and computer science to approximate complicated functions that are hard to compute directly. Mastering this core skill will make every subsequent topic in infinite sequences and series much easier to understand.

- [Lagrange Error Bound](https://www.owlsprep.com/study/ap-calculus-bc-u10-lagrange-error-bound/)
- [Radius and interval of convergence of power series](https://www.owlsprep.com/study/ap-calculus-bc-u10-radius-and-interval-of-convergence/)
- [Finding Taylor or Maclaurin series for a function](https://www.owlsprep.com/study/ap-calculus-bc-u10-finding-taylor-or-maclaurin-series/)

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