Polynomial, exponential, logarithmic, and trigonometric functions
Calculate the derivatives of polynomial, exponential, logarithmic, and trigonometric functions.
However, there are many engineering applications in which we need to take the derivative of a function that is too complex to be written in any of these forms.
You’ll apply these same ideas to calculate derivatives using two finite difference formulas; and you’ll also learn the importance of mesh size in determining the accuracy of these calculations.
To simplify, let’s assume that we’re working with a uniform one-dimensional mesh, with the distance between adjacent nodes being the “mesh size” h.
Knowing the value of a function f at each node in the mesh, your objective is to calculate the derivative of at node .
To derive the two formulas you’ll be using, we start with the definition of the derivative:
If we applied this formula to our grid values, we would get the forward difference expression
and the backward difference expression
Note that these are approximations to the value of the derivative, since we’re not taking the limit as h goes to zero; but we can improve the approximation by taking the average of these two difference formulas:
which simplifies to the centered difference expression
With this background, here’s your assignment:
- assume the function f is defined as f(x) = 5x4 – 9x3 + 2
- Use the power rule to find the derivative f’(x) and evaluate that derivative at x = 1.7. Note: To avoid round-off error, retain at least six decimal places in your calculations.
- Use the “forward difference” and “centered difference” formulas to estimate f’(x) at x = 1.7 for three different values of the mesh sizes
- h = 0.1
- h = 0.01
- h = 0.001
Note: To avoid round-off error, retain at least six decimal places in your functional evaluations, and retain the maximum possible number of decimal places in calculations of the forward and centered difference approximations.
- Use your calculated values to fill in this table:
Calculate derivatives using two finite difference formulas:
|h||forward difference approximation||centered difference approximation||exact derivative|
- Answer the following two questions:
- Which formula yields a better approximation: The forward difference or the centered difference?
- What effect does reducing the mesh size h have upon the accuracy of these approximations?
Be sure to show all of your work in making these calculations.