Linearization

考纲要求

• Understand how a graph of $y$ against $x$ is used to find constants $m$ and $c$ in $y=mx+c$
• Understand how $\lg y$ against $\lg x$ finds constants in $y=ax^n$
• Understand how $\ln y$ against $x$ finds constants in $y=ae^{kx}$
• Decide what derived quantities to calculate from raw data

常见题型

• Standard linearization — simple rearrangement
• Log-log linearization — power laws
• Log-linear linearization — exponential relationships
• Reciprocal linearization — inverse relationships

核心公式

• $y = mx + c$: plot $y$ vs $x$, gradient $= m$, intercept $= c$
• $y = ax^n$: plot $\lg y$ vs $\lg x$, gradient $= n$, intercept $= \lg a$
• $y = ae^{kx}$: plot $\ln y$ vs $x$, gradient $= k$, intercept $= \ln a$

常见错误

• Choosing wrong axes
• Forgetting units in logarithms: $\ln(V/\text{V})$, not $\ln V$
• Mixing $\ln$ and $\lg$