Track orders and small angles

Choose a strategy for an extended expression

The source gives three common strategies and explicitly says the list is not exhaustive.

Expression: $\cos x/x^2$

Strategy: Rewrite as a power of x times the standard cosine expansion.

Expression: $e^{1-2x}$

Strategy: Apply the law of indices, then replace the exponential argument by −2x.

Expression: $(1-x)^{-2}$

Strategy: Recognize the derivative of the geometric series.

The complete source illustrations are:

$$\frac{\cos x}{x^2}=\frac1{x^2}\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\right)=\frac1{x^2}-\frac12+\frac{x^2}{24}-\cdots.$$

This representation uses x≠0 and has a negative power: it is not a Maclaurin series of a function defined at zero.

$$e^{1-2x}=e\,e^{-2x}=e\left[1+(-2x)+\frac{(-2x)^2}{2!}+\cdots\right]=e(1-2x+2x^2-\cdots).$$ $$\frac1{(1-x)^2}=\frac d{dx}\left(\frac1{1-x}\right)=\frac d{dx}(1+x+x^2+\cdots)=1+2x+\cdots.$$

For the geometric series, termwise differentiation is used inside |x|<1. Self-Practice 2 applies the coefficient method and standard-series composition; its part (b) target is printed in the ask, rather than supplied as a separate worked answer.

Use exact angle identities before small-angle truncations

Example 9 and Self-Practice 3 concern an angle with a fixed offset. Small x does not make x+π/6 or π/4+x/3 a small angle. Apply the sine, cosine or tangent addition formula first, then approximate only the small increment in radians. The source’s special-angle values and both cosine addition identities are retained in the relevant working.

The closing remark distinguishes exact equality from approximation: use = for an exact transformation and ≈ when replacing an expression by a truncation. The square-area proof in Example 11 likewise uses exact tangent subtraction and exact triangle areas before its final approximation.

The complete printed geometry solution appears on Student PDF 26 and Teacher PDF 25; Student PDF 25 is an extra blank page. Their shared proof and callouts are integrated into the example, and the source square/angle diagram is reconstructed with a text alternative.

Track order after division and cancellation

Work backwards from the requested final power. If division removes a factor $x^m$, the numerator must be known $m$ powers further. Cancellation may expose terms that looked unnecessary earlier.

For example, obtaining a quotient through $x^2$ after cancelling $x^2$ can require fourth-degree terms in both numerator and denominator. Keep $x\ne0$ during that cancellation, then consider a limit separately if asked.

Example 10 rewrites a reciprocal denominator as a negative power and retains only needed terms. Part (a) has a denominator nonzero at zero and gives sinθ/cos2θ≈θ through quadratic order. Part (b) has a denominator vanishing quadratically.

Its printed answer 2 is retained as a leading approximation; to retain every quadratic term of the final quotient, fourth-degree terms must first be kept in the numerator and denominator. The independently derived refinement is $2+\tfrac56\theta^2+O(\theta^4)$ for small nonzero θ.

Self-Practice 4 keeps the positive square root and the √2 prefactor when normalizing $\sqrt{1+\cos\theta}$. Example 11 has an outer factor θ, so its reciprocal bracket is needed only through the linear term to obtain a²θ(1−θ).

The complete source review and closing practice map are retained with this chapter; the historical partial-review notes describe earlier ingestion stages.

The closing reading suggestion names The Manga Guide to Calculus by Hiroyuki Kojima; its cover also credits Shin Togami and Becom Co., Ltd. The source suggests borrowing it from the CJC Library, which is a printed historical suggestion rather than a check of current availability.

Its illustrated extract connects the local linear approximation $f(x)\approx f(a)+f^{\prime}(a)(x-a)$ near $a$ with higher-degree approximations. Maclaurin uses $a=0$. It also models compound interest: principal $M$ at annual rate $x$ grows to $M(1+x)^n$ after $n$ years.

For a positive integer $n$, the binomial expansion is a finite polynomial, with $\binom nr=n!/[r!(n-r)!]$; its first coefficients are $1,n,n(n-1)/2,n(n-1)(n-2)/6$. This finite case has no infinite-series convergence restriction. Original cover/comic artwork remains in the source evidence archive.

Review each of the eight skills at the end of lecture and again at the end of tutorial, then use the printed practice map below.

Plan the required order before dividing

Find the expansion of $(\sin x-x)/x^3$ through the term in $x^2$, for small nonzero x. Explain why keeping the sine expansion only through $x^3$ misses part of the requested answer. Distinguish the quotient at zero from its limit.

After your attempt: compare the reasoning

Division by $x^3$ lowers powers by three, so the numerator must be retained through $x^5$:

$\sin x=x-\frac{x^3}{6}+\frac{x^5}{120}+\cdots.$

Thus $(\sin x-x)/x^3=-1/6+x^2/120+\cdots$ for nonzero x near zero. A cubic truncation of sine supplies only the constant after division. The original quotient is undefined at zero, although its limit is $-1/6$. These are the first few terms, with no general-term derivation required.