Numerical Differentation and Integration

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4.1 Numerical Differentiation

• forward: \(f'(x)\approx\frac{f(x+h)-f(x)}{h}\)
• backward: \(f'(x)\approx\frac{f(x)-f(x-h)}{h}\)
• to calculate the differentiation of \(f(x)\) at \(x\), we need a polyminial \(P(x)\) that passes through the points(at least) \((x-h,f(x-h)),(x,f(x)),(x+h,f(x+h))\)
• Therefore, lagrange interpolation can be used
  

Problems might occur

4.3 Elements of Numerical Integration

• Approximate \(I=\int_{a}^{b}f(x)dx\)
• we might use the interpolated polyminial to replace \(f(x)\)