- Asked 3/13
- 2073 Magh · 8 marks
Fit curve y = a b^x (decay data)
Short answer
Take log10 of y = ab^x to get Y = A + Bx. Least squares gives B = -0.2563 and A = 1.0835, so a = 10^A = 12.1199 and b = 10^B = 0.5543. The fitted curve is y = 12.1199(0.5543)^x.
Fit the following set of data to a curve of the form y = ab^x.
| x | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
|---|---|---|---|---|---|---|---|
| y | 8.2 | 5.2 | 3.1 | 2.5 | 1.7 | 1.6 | 1.4 |
Take logarithms (base 10): y = a·bˣ → log y = log a + x·log b. Put Y = log y, A = log a, B = log b, so Y = A + Bx is a straight line in x.
| x | y | Y = log y | x² | xY |
|---|---|---|---|---|
| 1 | 8.2 | 0.9138 | 1.0000 | 0.9138 |
| 1.5 | 5.2 | 0.7160 | 2.2500 | 1.0740 |
| 2 | 3.1 | 0.4914 | 4.0000 | 0.9827 |
| 2.5 | 2.5 | 0.3979 | 6.2500 | 0.9949 |
| 3 | 1.7 | 0.2304 | 9.0000 | 0.6913 |
| 3.5 | 1.6 | 0.2041 | 12.2500 | 0.7144 |
| 4 | 1.4 | 0.1461 | 16.0000 | 0.5845 |
| Σx = 17.5 | 3.0998 | 50.7500 | 5.9557 |
Normal equations
ΣY = n·A + B·Σx → 3.0998 = 7A + 17.5000B
ΣxY = A·Σx + B·Σx² → 5.9557 = 17.5000A + 50.7500B
Solving: B = (nΣxY − ΣxΣY)/(nΣx² − (Σx)²) = -0.2563, A = (ΣY − BΣx)/n = 1.0835.
a = 10ᴬ = 12.1199, b = 10ᴮ = 0.5543.
Answer: y = 12.1199 · (0.5543)ˣ (decay curve, b < 1; at x = 2 it gives 3.72 against 3.1).
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Fit y = a b^x and find y(7) · Numerical Method, chapter 4 · asked 2×
- Fit y = ax/(b + x) · Numerical Method, chapter 4 · asked 2×
- Fit y = a b^x (growth data) · Numerical Method, chapter 4
- Fit curve y = a x^b (A) · Numerical Method, chapter 4 · asked 2×
- Fit y = a e^(bx) · Numerical Method, chapter 4
- Quadratic least squares fit · Numerical Method, chapter 4
- Fit y = ln(ax + b) · Numerical Method, chapter 4 · asked 2×
- Fit y = a + b√x · Numerical Method, chapter 4
- Least squares fit for calibration data · Metrology, chapter 2
- Rankine's data, 8° curve, 187.62 m long chord · Surveying II, chapter 6
- Demagnetizing via B-H curve; data storage material · Electrical Engineering Material, chapter 4
- Presentation of precipitation data, IDF curve · Engineering Hydrology, chapter 2