Mata Kuliah : Metode Numerik�Minggu ke 14

Mahasiswa dapat melakukan komputasi integrasi numerik dengan metode Trapezoidal, Simpson 1/3

Tujuan perkuliahan

Defferentiation and Integration

Type of Function

• Discrete�Forward Difference, Direct Fit Polynomial, Lagrange Method
• Continues�Forward Difference, Backward Difference, Central Difference, and Higher order from Taylor series.�

Example

• Discrete

The upward velocity of a rocket is given as a function of time in Table 1.

Using forward divided difference, find the acceleration of the rocket at .

Table 1 Velocity as a function of time

Example

Example

• Continuous

The velocity of a rocket is given by

where

is given in m/s and

is given in seconds.

1. Use forward difference approximation of the first derivative of to calculate the acceleration at . Use a step size of .

Example

Hence

Integration

Trapezoidal Rule

Example

The vertical distance covered by a rocket from t=8 to t=30 seconds is given by:

1. Use single segment Trapezoidal rule to find the distance covered.
2. Find the true error, for part (a).
3. Find the absolute relative true error, for part (a).

Example

a)

Example

c)

Example�(How about Double Segment?)

Multiple Segment (visualization)

Divide into equal segments as shown in Figure 4. Then the width of each segment is:

The integral I is:

Comparation

Table 1 gives the values obtained using multiple segment Trapezoidal rule for

Table 1: Multiple Segment Trapezoidal Rule Values

Exact Value=11061 m

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