PERTEMUAN 4
Differensial ( continue)
Project II.
Investigate the numerical differentiation formulae
and truncation error bound
where
. The truncation error is investigated. The round off
error from computer arithmetic using computer numbers will be studied in another
module.
Enter the formula for numerical differentiation.
Aside. It looks like the formula is a second divided difference, i.e. the difference
quotient of two difference quotients. Such is the case.
Aside. From a mathematical standpoint, we expect that the limit of the second divided
difference is the second derivative. Such is the case.
Example 5. Consider the function
. Find the formula for the fourth
derivative
, it will be used in our explorations for the remainder term and the
truncation error bound. Graph
. Find the bound
. Look
at it's graph and estimate the value
, be sure to take the absolute value if necessary.
Solution 5.
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Example 6 (a). Compute numerical approximations for the derivatives
, using step sizes
.
6 (b). Plot the numerical approximation
over the interval
. Compare it with the graph of
over the interval
.
Solution 6 (a).
Example 6 (a). Compute numerical approximations for the derivatives
, using step sizes
.
Solution 6 (a).
Solution 6 (b).
Example 6 (b). Plot the numerical approximation
. Compare it with the graph of
over the interval
over the interval
.
Solution 6 (b).
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Example 7. Plot the absolute error
over the interval
, and estimate the maximum absolute error over the interval.
7 (a). Compute the error bound
and observe
that
over
.
7 (b). Since the function f[x] and its derivative is well known, and we have the graph for
, we can observe that the maximum error on the given interval
occurs at x=0. Thus we can do better that "theory", we see that
over
.
Solution 7.
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Example 8. Investigate the behavior of
of
then the error bound is reduced by
Solution 8.
. If the step size is reduced by a factor
. This is the
behavior.
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Example 11. Given
, find numerical approximations to the
derivative
, using two points and the central difference formula.
Solution 11.
Example 11. Given
, find numerical approximations to the
derivative
, using two points and the central difference formula.
Solution 11.
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Animation 6. ( Numerical Differentiation Numerical Differentiation ).
Example 12. Given
, find numerical approximations to the
derivative
, using three points and the forward difference formula.
Solution 12.
Example 12. Given
, find numerical approximations to the
derivative
, using three points and the forward difference formula.
Solution 12.
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Animation 2. ( Numerical Differentiation Numerical Differentiation ).
Example 13. Given
, find numerical approximations to the
derivative
, using three points and the backward difference formula.
Solution 13.
Example 14. Given
, find numerical approximations to the
derivative
, using three points and the central difference formula.
Solution 14.
Example 15. Given
, find numerical approximations to the second
derivative
, using three points and the forward difference formula.
Solution 15.
Example 16. Given
, find numerical approximations to the second
derivative
, using three points and the backward difference formula.
Solution 16.
Example 17. Given
, find numerical approximations to the second
derivative
, using three points and the central difference formula.
Solution 17.
Animations (Numerical Differentiation Numerical Differentiation). Internet
hyperlinks to animations.
Old Lab Project (Numerical Differentiation Numerical Differentiation). Internet
hyperlinks to an old lab project.
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Research Experience for Undergraduates
Numerical Differentiation Numerical Differentiation Internet hyperlinks to web sites
and a bibliography of articles.
Download this Mathematica Notebook Numerical Differentiation
Maclaurin and Taylor Polynomials Module
Lagrange Polynomials Module
Newton Polynomials Module
Chebyshev Polynomials Module
Pade Approximations Module
Terima Kasih
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