Finding Taylor Polynomial Approximations of Functions

3. Visual Connection

5. Concept Development

BC-only topic. A Taylor polynomial replaces a sufficiently differentiable function by a polynomial that has the same local value, slope, concavity, and higher-derivative behavior at a chosen center.

The degree-n Taylor polynomial

For a function \(f\) centered at \(x=a\),

In sigma notation,

Why the coefficients contain factorials

If \(P_n(x)=c_0+c_1(x-a)+\cdots+c_n(x-a)^n\), then evaluating its \(k\)th derivative at the center removes every other power and gives

To force \(P_n^{(k)}(a)=f^{(k)}(a)\), the coefficient must be \(c_k=f^{(k)}(a)/k!\).

Derivative-matching property

The degree-\(1\) Taylor polynomial is the familiar tangent-line approximation. Degree \(2\) also matches concavity, and each higher degree captures one more derivative at the center.

Taylor versus Maclaurin

• Taylor polynomial: centered at any specified \(a\), so powers are \((x-a)^k\).
• Maclaurin polynomial: the special case \(a=0\), so powers are \(x^k\).

Maclaurin is not a different formula; it is a Taylor polynomial centered at zero.

A construction workflow

1. Identify the center \(a\) and requested degree \(n\).
2. Find \(f(a),f'(a),\ldots,f^{(n)}(a)\).
3. Place each value over its matching factorial.
4. Use powers of \(x-a\), not powers of \(x\), unless \(a=0\).
5. Retain zero coefficients when they clarify the requested degree or derivative pattern.
6. Simplify only after the structure is correct.

Using the polynomial to approximate a value

For an input \(x=a+h\) near the center, substitute into the polynomial:

The approximation is usually better when \(|h|\) is small and when an appropriate higher degree is used. Topic 10.12 supplies a guaranteed Lagrange error bound.

Approximation is not automatic equality

Write \(f(x)\approx P_n(x)\) unless \(f\) itself is a polynomial of degree at most \(n\) or an exact equality has been established. Matching finitely many derivatives at one point does not make two functions identical everywhere.

Degree versus number of nonzero terms

A degree-\(n\) Taylor polynomial includes powers through \((x-a)^n\), but some coefficients may be zero. For example, the degree-\(5\) Maclaurin polynomial for \(\sin x\) has only three nonzero terms.

Reading derivative data from a table

If a table supplies \(f^{(k)}(a)\), insert those values directly. Do not differentiate the table entries or treat \(f^{(k)}(a)\) as the coefficient before dividing by \(k!\).

Recovering derivatives from a polynomial

If a Taylor polynomial centered at \(a\) contains \(c_k(x-a)^k\), then

A missing power has coefficient zero, so the corresponding derivative at the center is zero.

Finding derivatives from a differential equation

When \(f\) is defined by an initial-value problem, use the equation to find \(f'(a)\), differentiate the equation to obtain \(f''(a)\), and continue recursively. Substitute the initial condition after each derivative formula is valid.

Choosing a useful center

A center close to the target input keeps powers of \(h=x-a\) small. Centers where function and derivative values are simple, such as \(0\), \(1\), or a special trigonometric angle, also reduce arithmetic.

AP-style checklist

1. State the center and degree.
2. Show the needed derivative values at the center.
3. Write the polynomial explicitly with factorial denominators.
4. Use the correct shifted power in every term.
5. For a numerical approximation, substitute the requested input into the displayed polynomial.
6. Use approximation notation and round only at the end.

6. Detailed Worked Example and Error Check

Example 1: Exponential Maclaurin polynomial

Every derivative of \(e^x\) equals \(e^x\), so every derivative value at \(0\) is \(1\). Thus

For example, \(e^{0.2}\approx P_4(0.2)\approx1.2214\).

Example 2: Sine and zero coefficients

The derivatives of \(\sin x\) cycle through \(\sin x,\cos x,-\sin x,-\cos x\). At \(0\), the values cycle \(0,1,0,-1\). The degree-\(5\) polynomial is

Although only three terms are nonzero, the highest power is \(5\).

Example 3: Cosine Maclaurin polynomial

Using the cosine derivative cycle,

The odd-power coefficients vanish because every odd derivative of \(\cos x\) is zero at the origin.

Example 4: Logarithm centered at 1

For \(f(x)=\ln x\), the first derivative values at \(1\) are \(1,-1,2\). Hence

At \(x=1.2\), \(\ln(1.2)\approx0.2-0.02+0.008/3\approx0.182667\).

Example 5: Square root centered at 4

For \(f(x)=\sqrt x\),

Therefore

This gives \(\sqrt{4.1}\approx P_2(4.1)=2.02484375\).

Example 6: Construct from a derivative table

Suppose \(f(2)=3\), \(f'(2)=-1\), \(f''(2)=4\), and \(f'''(2)=6\). Then

Thus \(f(2.1)\approx3-0.1+2(0.1)^2+(0.1)^3=2.921\).

Example 7: Recover derivative values

Suppose the fourth-degree Taylor polynomial centered at \(-1\) is

Then \(f(-1)=2\), \(f'(-1)=-3\), \(f''(-1)=0\), \(f'''(-1)=3!(5)=30\), and \(f^{(4)}(-1)=0\).

Example 8: Initial-value problem

Let \(y'=xy\) and \(y(0)=2\). Successive differentiation gives

Therefore the degree-\(4\) Maclaurin polynomial is

Example 9: The tangent line is P1

If \(f(3)=7\) and \(f'(3)=-2\), then

This is exactly the linearization of \(f\) at \(x=3\).

Example 10: Exactness for a polynomial

If \(f(x)=x^3-2x+4\), its degree-\(3\) Taylor polynomial about any center equals \(f(x)\) after expansion. All derivatives above the third are zero, so no remainder remains.

Common errors

• Forgetting the factorial in a coefficient.
• Using \(x^k\) instead of \((x-a)^k\) at a nonzero center.
• Substituting the target input as the center.
• Confusing degree \(n\) with \(n\) nonzero terms.
• Writing \(f(x)=P_n(x)\) when only an approximation is justified.
• Using \(f^{(k)}(a)\) directly as the coefficient instead of dividing by \(k!\).
• Dropping a zero coefficient and then misidentifying the requested degree.
• Reading \(c_k\) as \(f^{(k)}(a)\) when extracting derivatives.
• Rounding derivative values before constructing the polynomial.
• Using an error-bound formula from Topic 10.12 without checking its hypotheses.
• Giving only a numerical approximation when the polynomial itself is requested.