# 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.

Thus,

And

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) = 5x
^{4}– 9x^{3}+ 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 |

0.1 | |||

0.01 | |||

0.001 |

- 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.__