Sampling of curvesbased on various linear and non-linear function :Indian Economic Service

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Sampling of Curves Based on Various Linear and Non-Linear Functions

1. Introduction

📌 Sampling of curves refers to the process of selecting points from a function to analyze its behavior, estimate parameters, and make predictions.
📌 Curves can be classified as linear or non-linear, depending on their mathematical expressions.
📌 Used in economics, physics, statistics, and machine learning for data modeling and prediction.

✔ Example: In economics, demand and supply curves are often estimated using samples from real-world data.

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2. Linear Functions and Their Sampling

🔹 (1) Definition

✔ A linear function is of the form: y=mx+cy = mx + c

where:

• yy = dependent variable,
• xx = independent variable,
• mm = slope (rate of change),
• cc = intercept (value of yy when x=0x = 0).

✔ Example: A company’s revenue (yy) increases at a fixed rate (mm) with respect to sales (xx).

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🔹 (2) Sampling from a Linear Function

✔ Equidistant Sampling: Select equally spaced xx-values.
✔ Random Sampling: Select xx-values randomly within a range.
✔ Stratified Sampling: Divide xx into intervals and select points from each.

✔ Example: Sampling from y=2x+3y = 2x + 3 at x=1,2,3,4,5x = 1, 2, 3, 4, 5: y1=2(1)+3=5,y2=2(2)+3=7,y3=2(3)+3=9,…y_1 = 2(1) + 3 = 5, \quad y_2 = 2(2) + 3 = 7, \quad y_3 = 2(3) + 3 = 9, \dots

📌 This forms a straight-line data set: (1,5), (2,7), (3,9), (4,11), (5,13).

✔ Application: Used in trend analysis, cost estimation, and demand prediction.

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3. Non-Linear Functions and Their Sampling

🔹 (1) Quadratic Functions

✔ Equation: y=ax2+bx+cy = ax^2 + bx + c

✔ Example: Production functions, where output increases at a diminishing rate.
✔ Sampling: Choose xx values and compute yy.

Example Sampling from y=x2−2x+1y = x^2 – 2x + 1:
✔ x=−2,−1,0,1,2x = -2, -1, 0, 1, 2 gives points:
(4, 3, 1, 1, 3) → Parabolic curve.

📌 Application: Cost curves in economics (Total Cost vs. Output).

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🔹 (2) Exponential Functions

✔ Equation: y=aebxy = a e^{bx}

✔ Example: Economic growth models, compound interest, population growth.
✔ Sampling: Choose xx values and compute yy.

✔ Example Sampling from y=2e0.5xy = 2e^{0.5x}:

• x=0,1,2,3x = 0, 1, 2, 3 → y=2,3.3,5.4,9.1y = 2, 3.3, 5.4, 9.1
  📌 Application: Inflation, investment growth.

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🔹 (3) Logarithmic Functions

✔ Equation: y=alog⁡(bx)y = a \log (bx)

✔ Example: Diminishing returns in production.
✔ Sampling: Choose xx, calculate yy.

✔ Example Sampling from y=3log⁡(2x)y = 3\log(2x):

• x=1,2,4,8x = 1, 2, 4, 8 → y=0,2,4,6y = 0, 2, 4, 6.
  📌 Application: Learning curves, wage determination.

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🔹 (4) Power Functions

✔ Equation: y=axby = ax^b

✔ Example: Cobb-Douglas Production Function: Q=ALαKβQ = A L^\alpha K^\beta

✔ Sampling: Choose values of L,KL, K, compute QQ.

✔ Example: If Q=10L0.5K0.3Q = 10 L^{0.5} K^{0.3}, then:

• L=1,K=2L = 1, K = 2 → Q=10(10.5)(20.3)Q = 10(1^{0.5})(2^{0.3}).
  📌 Application: Used in production and elasticity analysis.

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4. Curve Fitting and Estimation from Samples

✔ Interpolation: Estimate values within the sampled data range.
✔ Regression Analysis: Fit linear/non-linear models to data.
✔ Machine Learning: Use sampled points to train models.

📌 Example: Estimating demand curve Q=f(P)Q = f(P) using regression.

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5. Conclusion

✔ Linear and non-linear functions have distinct sampling methods.
✔ Linear sampling is simple, while non-linear sampling follows complex patterns.
✔ Applications include economics, finance, engineering, and AI.