Linear Law (straight-line Form)
Muhtasari
Experimental data rarely falls on a straight line, but a clever change of variables can often make it do so. By taking logarithms, a power law or an exponential law can be straightened into the form \(Y=mX+c\). Then the gradient and intercept of the straight-line graph hand you the unknown constants.
In this lesson you will convert relationships such as \(y=Ax^n\) and \(y=Ab^x\) into straight-line form by taking logarithms, identify what to plot, and read off the constants from the gradient and intercept. This is the bridge between curved real-world data and the simple line you know how to analyse.
Malengo
- Transform given relationships, such as y = Ax to the power n and y = Ab to the power x, to and from straight-line form.
- Determine unknown constants by calculating the gradient or intercept of the transformed straight-line graph.
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Tathmini ya Somo
Hongera kwa kukamilisha somo la Linear Law (straight-line Form). Sasa kwa kuwa umechunguza dhana na mawazo muhimu, ni wakati wa kuweka ujuzi wako kwa mtihani. Sehemu hii inatoa mazoezi mbalimbali maswali yaliyoundwa ili kuimarisha uelewaji wako na kukusaidia kupima ufahamu wako wa nyenzo.
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- To straighten y = Ax^n, you plot: A. y against x B. lg y against lg x C. lg y against x D. y against lg x Answer: B
- To straighten y = Ab^x, you plot: A. lg y against lg x B. lg y against x C. y against x D. y against lg x Answer: B
- Taking logs of y = Ax^n gives: A. lg y = n lg x + lg A B. lg y = A lg x + n C. lg y = lg x + n lg A D. y = n lg x + A Answer: A
- If lg y against lg x has gradient 3 and intercept 0.7 for y = Ax^n, then n is: A. 0.7 B. 3 C. 10^0.7 D. 0.3 Answer: B
- If the intercept of the straight-line graph is lg A = 1, then A equals: A. 1 B. 10 C. 0.1 D. 100 Answer: B
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Vitabu Vinavyopendekezwa
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