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Taylor Polynomial
Approximations

Learn how to approximate any function using polynomial expressions built from derivatives at a single point.

AP Calculus BC

What is a Taylor Polynomial?

A Taylor polynomial is a polynomial that approximates a function near a specific point \( x = a \). It is constructed so that the polynomial and the original function share the same value and the same first several derivatives at that point.

The Big Idea: Match Derivatives

Why Do They Matter?

Computation
Calculators use Taylor polynomials to evaluate \( \sin x \), \( \cos x \), \( e^x \), and more.
Physics
Small-angle approximations like \( \sin\theta \approx \theta \) are first-degree Taylor polynomials.
Error Bounds
The Lagrange error bound lets you quantify how accurate the approximation is.

The General Formula

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

Here \( a \) is the center of the polynomial, \( f^{(k)}(a) \) is the \(k\)-th derivative evaluated at \( a \), and \( n \) is the degree of the approximation.

Taylor vs. Maclaurin

Taylor Polynomial
Centered at any point \( x = a \). Formula uses \( (x - a)^k \).
Maclaurin Polynomial
Special case where \( a = 0 \). Formula simplifies to \( x^k \) terms.

Polynomial vs. Series

Taylor Polynomial Taylor Series
Terms Finitely many (\(n+1\)) Infinitely many
Notation \( P_n(x) \) \( \displaystyle\sum_{k=0}^{\infty} \)
Result A polynomial (exact degree \(n\)) A power series (may converge)
AP Focus Finding & using approximations Convergence & interval of convergence

How to Build a Taylor Polynomial

1
Identify the center \( a \) and the desired degree \( n \).
2
Compute derivatives \( f(a),\; f'(a),\; f''(a),\; \ldots,\; f^{(n)}(a) \).
3
Plug into the formula: each term is \( \dfrac{f^{(k)}(a)}{k!}(x-a)^k \).
4
Simplify the resulting polynomial expression.

Example 1: \( e^x \) about \( a = 0 \)

Find the 4th-degree Maclaurin polynomial for \( f(x) = e^x \).

1
All derivatives of \( e^x \) are \( e^x \). At \( x=0 \): \( f^{(k)}(0) = 1 \) for every \( k \).
2
Each coefficient is \( \dfrac{1}{k!} \).
3
Result: \( P_4(x) = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \dfrac{x^4}{24} \).

Example 2: \( \sin x \) about \( a = 0 \)

Find the 5th-degree Maclaurin polynomial for \( f(x) = \sin x \).

\(k\) \(f^{(k)}(x)\) \(f^{(k)}(0)\) Term
0\(\sin x\)00
1\(\cos x\)1\(x\)
2\(-\sin x\)00
3\(-\cos x\)\(-1\)\(-\dfrac{x^3}{6}\)
4\(\sin x\)00
5\(\cos x\)1\(\dfrac{x^5}{120}\)

Visualizing \( \sin x \) Approximations

Example 3: \( \ln x \) about \( a = 1 \)

Find the 3rd-degree Taylor polynomial for \( f(x) = \ln x \) centered at \( a = 1 \).

1
Derivatives: \( f(x)=\ln x \), \( f'(x)=\frac{1}{x} \), \( f''(x)=-\frac{1}{x^2} \), \( f'''(x)=\frac{2}{x^3} \).
2
At \( x = 1 \): \( f(1)=0 \), \( f'(1)=1 \), \( f''(1)=-1 \), \( f'''(1)=2 \).
3
Result: \( P_3(x) = (x-1) - \dfrac{(x-1)^2}{2} + \dfrac{(x-1)^3}{3} \).

Visualizing \( \ln x \) near \( a = 1 \)

Essential Maclaurin Polynomials

Function Maclaurin Polynomial (first terms)
\( e^x \)\( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
\( \sin x \)\( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)
\( \cos x \)\( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \)
\( \frac{1}{1-x} \)\( 1 + x + x^2 + x^3 + \cdots \)
\( \ln(1+x) \)\( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)

Common Mistakes to Avoid

✕ Forgetting the factorial
Each term divides by \( k! \). Writing \( f''(a)(x-a)^2 \) instead of \( \frac{f''(a)}{2!}(x-a)^2 \) is the most common error.
✕ Wrong center
If centered at \( a=1 \), every power must be \( (x-1)^k \), not \( x^k \). Evaluate derivatives at \( x=1 \), not \( x=0 \).
✕ Losing signs
Track negative derivatives carefully. For \( \cos x \): \( f''(0) = -1 \), so the \( x^2 \) term is \( -\frac{x^2}{2} \), not \( +\frac{x^2}{2} \).

Question 1 of 5

What is the second-degree Maclaurin polynomial for \( f(x) = e^x \)?

Question 2 of 5

The third-degree Taylor polynomial for \( f(x) = \ln x \) centered at \( a = 1 \) is:

Question 3 of 5

In the Maclaurin polynomial for \( \sin x \), which powers of \( x \) appear?

Question 4 of 5

If \( f(x) = \cos x \), what is \( \dfrac{f''(0)}{2!} \)?

Question 5 of 5

What is the second-degree Taylor polynomial for \( f(x) = \sqrt{x} \) centered at \( a = 4 \)?