📗 Derivatives can be approximated using Newton's formula, \(\dfrac{d f}{d x} \approx \dfrac{f\left(x + h\right) - f\left(x\right)}{h}\), for some small \(h\) close to \(0\).
➩ The derivative of \(f\) is usually approximated by \(\dfrac{d f}{d x} \approx \dfrac{f\left(x + h\right) - f\left(x - h\right)}{2 h}\).
➩ There are more accurate approximations using finite differences, for example, \(\dfrac{d f}{d x} \approx \dfrac{-\left(f\left(x + 2 h\right)\right) + 8 f\left(x + h\right) - 8 f\left(x - h\right) + f\left(x - 2 h\right)}{12 h}\).
📗 Partial derivatives of a multivariate function are derivatives with respect to one of the variables holding the other variables constant: \(\dfrac{\partial f\left(x, y\right)}{\partial x} \approx \dfrac{f\left(x + h, y\right) - f\left(x - h, y\right)}{2 h}\) and \(\dfrac{\partial f\left(x, y\right)}{\partial y} \approx \dfrac{f\left(x, y + h\right) - f\left(x, y - h\right)}{2 h}\).
➩ The gradient of a multivariate function is hte vector of partial derivatives, one for each variable, \(\nabla f\left(x, y\right) = \begin{bmatrix} \dfrac{\partial f}{\partial x} \\ \dfrac{\partial f}{\partial y} \end{bmatrix}\).
📗 Definite integrals can be approximated using finite Riemann sums.
➩ If the right Riemann sum is used, then \(\displaystyle\int_{x_{0}}^{x_{1}} f\left(x\right) d x \approx \displaystyle\sum_{i=1}^{n} f\left(x_{0} + i \cdot h\right) h\), for \(h = \dfrac{x_{1} - x_{0}}{n}\), for some large \(n\).
➩ The left Riemann sum (replace \(i\) by \(i - 1\)) can also be used.
➩ The mid-point rule can be used too (replace \(i\) by \(i - 0.5\)).
➩ The points at which the function is evaluated can be chosen adaptively or randomly.
📗 It is possible to compute the numerical integral as a function. Instead of having the \(+C\), an arbitrary constant can be used.
➩ function d = intf(f)
➩ h = 0.01;
➩ d = @(x)(sum(f(h:h:x)) * h)
➩ end
📗 integral(f, x0, x1) finds the numerical definite integral \(\displaystyle\int_{x_{0}}^{x_{1}} f\left(x\right) d x\).
📗 int(f) finds the indefinite integral of \(f\) and returns a function. It requires MATLAB's Symbolic Math Toolbox.
📗 Notes and code adapted from the course taught by Professors Beck Hasti and Michael O'Neill.
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