Using a series to locate a different intersection

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Before you start

Maclaurin series. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

The curves are y = ln(1 + $x^{2}$ ) and y = 3x⁴. They meet at the origin and at points on either side.

Part a

Find the Maclaurin approximation to ln(1 + $x^{2}$ ) through $x^{4}$ .

Part b

Use this approximation to find $x^{2}$ at a nonzero intersection.

Part c

Explain why this is an approximation, and suggest a way to check its quality.

Model solution and review criteria

Part a

Substitute u = $x^{2}$ into ln(1+u) = u − $u^{2}$ /2 + …, obtaining $x^{2}$ − $x^{4}$ /2.

Substitute the whole $x^{2}$ argument.
Retain terms through $x^{4}$ .

Part b

Set $x^{2}$ − $x^{4}$ /2 = 3x⁴. For x ≠ 0, divide by $x^{2}$ : $x^{2}$ = $\frac{2}{7}$ .

Form the approximate intersection equation.
Exclude x = 0 before dividing.
Keep the symmetric positive and negative roots in interpretation.

Part c

The logarithm series was truncated. Evaluate ln(1+ $x^{2}$ ) − 3x⁴ at x = ±√( $\frac{2}{7}$ ) or include a further series term; the exact curves need not meet at the predicted points.

Identify the omitted higher-order terms.
Suggest a residual or a higher-order comparison.

Compare your own reasoning. No automatic examination marks are awarded.