Calculus II · Unit 4B · lesson

Taylor Polynomials

Concept

Learning objectives

construct a Taylor polynomial from derivative data and explain derivative matching at the center.

Taylor Polynomials

Explanation

A polynomial can be forced to imitate local behavior

A tangent line matches a function value and first derivative at one point. A quadratic Taylor polynomial also matches the second derivative. A degree- $n$ Taylor polynomial matches every derivative through order $n$ at the center. The coefficients are therefore determined, not guessed.

Derivative matching explains why Taylor polynomials are locally accurate. Near the center, the first unmatched derivative controls the leading error. Increasing degree does not automatically guarantee good behavior far away, but it incorporates more local information and often enlarges the useful interval.

Bridge

Match local derivative data one layer at a time

A Taylor polynomial is built to share a function's value, slope, curvature, and higher derivatives at one center. The constant term matches the value; the linear term matches the slope; the quadratic term repairs curvature; each additional degree matches one more derivative.

The factorial denominator is not an arbitrary convention. Differentiating $(x-a)^n$ exactly $n$ times produces $n!$ , so dividing by $n!$ makes the coefficient reproduce $f^{(n)}(a)$ . The polynomial is a local model whose useful range depends on the remainder, not merely on its degree.

**Higher Taylor degree matches more local shape**
A function near the center is compared with constant, tangent-line, quadratic, and cubic Taylor approximations. Each added degree follows the function farther from the center.
Higher Taylor degree matches more local shape. Function and degree 0, 1, 2, and 3 Taylor models near a common center.

Proof idea

The factorial makes derivative matching exact

The term

\frac{f^{(n)}(a)}{n!}(x-a)^n

vanishes under fewer than $n$ derivatives at $x=a$ , and after $n$ derivatives contributes exactly $f^{(n)}(a)$ . This triangular structure determines every coefficient.

**Taylor polynomial approximations**
Increasing the degree makes the polynomial agree with more local derivative data and typically follow the target function over a wider interval.
Taylor polynomial approximations. Function and degree 0, 1, 2, and 3 Taylor models near a common center.

How to read the visual

All approximations agree at $x=0$ . The higher-degree polynomials bend with the sine curve longer before the omitted terms become noticeable.

Concept

Taylor polynomial

The degree- $n$ Taylor polynomial for $f$ centered at $a$ is

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

Guided walkthrough

A cubic approximation for $e^x$

Every derivative of $e^x$ equals $e^x$ , so at $a=0$ each derivative value is $1$ . Thus

P_3(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}.

At zero, this polynomial matches $e^x$ and its first three derivatives.

Worked example

Build a polynomial directly from derivative data

Suppose

f(1)=2,\quad f'(1)=-3,\quad f''(1)=4,\quad f^{(3)}(1)=6.

The degree-three Taylor polynomial centered at $1$ is

T_3(x)=2-3(x-1)+\frac{4}{2!}(x-1)^2+\frac{6}{3!}(x-1)^3.

Thus

T_3(x)=2-3(x-1)+2(x-1)^2+(x-1)^3.

Each derivative condition can be checked by differentiating this polynomial.

Common mistake

Center the powers at a, not at zero unless a is zero

A Taylor polynomial about $a$ uses $(x-a)^n$ . Using $x^n$ instead generally matches derivatives at the wrong point.

Interactive checku4b-taylor_polynomials-01

Find the cubic Maclaurin polynomial for $e^x$ .

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Show hint

All derivatives at zero equal one.

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Exercise

Find the quadratic Taylor polynomial for $\ln x$ at $a=1$ .

Exercise

Find the fourth-degree Maclaurin polynomial for $\cos x$ .

Exercise

Verify derivative matching for the cubic exponential polynomial.

Exercise

Explain why factorials appear in the coefficients.

After the explanation

Use the section idea

Reading lens

Match value and derivatives at one center, then separate the polynomial approximation from the infinite-series convergence claim.

Mental model

Taylor coefficients encode local derivative data as a polynomial of increasing degree.

Decision

Choose the center, compute the derivative pattern, divide by factorials, and state whether you need a polynomial or an infinite series.

Common trap

Assuming every smooth-looking function equals its Taylor series everywhere.

Check yourself

Can you verify the first coefficients directly from derivatives at the center?

Source & rights

Original instruction with traceable references.

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