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Mahasiswa dapat melakukan komputasi diferensiasi numerik dengan High Order Differentiation

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Numerical Differentiation

Calculus is the mathematics of change. Because engineers and scientists must continuously deal with systems and processes that change, calculus is an essential tool of our profession. Standing at the heart of calculus is the mathematical concept of differentiation.

Numerical Differentiation

According to the dictionary definition, to differentiate means "to mark off by differences; distinguish;.. to perceive the difference in or between." Mathematically, the derivative, which serves as the fundamental vehicle for differentiation, represents the rate of change of a dependent variable with respect to an independent variable. As depicted in Fig. 21.1, the mathematical definition of the derivative begins with a difference approximation:

Numerical Differentiation

where y and f(x) are alternative representatives for the dependent variable and x is the independent variable. If Ar is allowed to approach zero, as occurs in moving from Fig. 21.la 10 c, the difference becomes a derivative:

Numerical Differentiation

The graphical definition of a derivative: as ∆r approaches zero in going from (a) to (c), the difference approximation becomes a derivative.

Numerical Differentiation

where dy/dx [which can also be designated as y' or f'(x;)]' is the first derivative of y with respect to r evaluated at x; As seen in the visual depiction of Fig. 21.Ic, the derivative is the slope of the tangent to the curve at x;

The second derivative represents the derivative of the first derivative,

Numerical Differentiation

Thus, the second derivative tells us how fast the slope is changing. It is commonly referred to as the curvature, because a high value for the second derivative means high curvature.

Finally, partial derivatives are used for functions that depend on more than one variable. Partial derivatives can be thought of as taking the derivative of the function at a point with all but one variable held constant. For example, given a function f that depends on both x and y, the partial derivative of f with respect to x at an arbitrary point (x, y) is defined as

Numerical Differentiation

Similarly, the partial derivative of f with respect to y is defined as

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To get an intuitive grasp of partial derivatives, recognize that a function that depends on two variables is a surface rather than a curve. Suppose you are mountain climbing and have access to a function f that yields elevation as a function of longitude (the east-west oriented

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Numerical Differentiation

Similarly, the partial derivative of f with respect to y is defined

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To get an intuitive grasp of partial derivatives, recognize that a function that depends on two variables is a surface rather than a curve. Suppose you are mountain climbing and have access to a function f that yields elevation as a function of longitude (the east-west oriented x axis) and latitude (the north-south oriented y axis).

Numerical Differentiation

Numerical Differentiation

Numerical Differentiation

Numerical Differentiation

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