考前速记

Must-Know Linearization Formulas

原始关系	线性化形式	Gradient	Y-intercept	反解常数
$y = mx + c$	$y$ vs $x$	$m$	$c$	—
$y = ax^n$	$\lg y$ vs $\lg x$	$n$	$\lg a$	$a = 10^{\text{int}}$
$y = ae^{kx}$	$\ln y$ vs $x$	$k$	$\ln a$	$a = e^{\text{int}}$
$y = a + \frac{b}{x}$	$y$ vs $\frac{1}{x}$	$b$	$a$	—
$y = \frac{a}{x}$	$y$ vs $\frac{1}{x}$	$a$	$0$	—
$y^2 = ax$	$y^2$ vs $x$	$a$	$0$	—

Problem Patterns to Recognise

看到...	立即想到...
$y = a x^n$	$\lg y = \lg a + n \lg x$
$y = a e^{kx}$	$\ln y = \ln a + kx$
$e^{-t/RC}$	gradient $= -1/RC$
$(R_1 + R_2)$ in denominator	plot $1/I$ against $(R_1 + R_2)$
$\frac{h_i}{h_o}$ vs $d$	gradient $= 1/f$
$\eta = H e^{E/kT}$	x-axis is $1/T$, NOT $T$

Red Flags

信号	可能的问题
公式中有 $\lg$ 或 $\ln$	确保对数参数无量纲
两个都是变量且关系为 power	用 $\lg$，不是 $\ln$
指数中有负号	gradient 有负号
题目给 $n$ 或 $k$ 要求计算	注意反解时是 $10^{\text{int}}$ 还是 $e^{\text{int}}$
常数有多个字母（如 $SZ$）	intercept 是 $\lg(SZ)$，不是分开的
参数有分数指数	试试 $\lg$ 或 reciprocal

Quick Checklist — Q2(a) Linearization

• Rearranged to $y = mx + c$ form
• $y$-axis quantity identified
• $x$-axis quantity identified
• Gradient expression correct (including sign)
• Y-intercept expression correct
• If $\lg$ or $\ln$: argument is dimensionless ($V/\text{V}$, not $V$)
• If $\lg$: $a = 10^{\text{intercept}}$
• If $\ln$: $a = e^{\text{intercept}}$

Quick Checklist — Q2(b) Graph & Calculation

• Table calculated with correct derived quantities
• Axes labelled with quantity and unit
• Sensible scales (at least half the grid used)
• Data points plotted correctly (small crosses)
• Line of best fit drawn (thin, straight, even distribution)
• Gradient calculated using two points far apart on the line
• Y-intercept read from graph where line crosses $y$-axis
• Constant calculated from gradient/intercept

提示

Paper 5 Q2 中，推导 gradient 和 intercept 表达式只占 2 分，但与后续的表格计算、画图、求常数紧密相关。公式推导错误会导致后面全线失分。仔细检查符号和对数底数。