| 1. | a) |
Compute the area under the curve f(x) = x2 + 5x + 2
between the
limits x = 0 and x = 2, using:
| i) | the trapezoidal rule with h = 0.2. |
| ii) | the Simpson's 1/3 rule with h = 0.2. |
|
| b) |
Which estimate is more accurate; the answer from part (a)-i) or (a)-ii) ?
Explain why. |
| 2. |
What is a step-size, and how is it determined, given data points
(x1,y1), (x2,y2),
(x3,y3) etc... ? |
| 3. |
Compute the derivative of the function f(x) =
4x2 + 6x - 2 using
the following approximations, with x = 3 and step sizes of 0.2,
0.1, and 0.05: |
| a) |
| i) | Forward Difference |
| ii) | What is the relationship between the step-size and the
accuracy of the estimate? |
|
| b) |
| i) | Backward Difference |
| ii) | What is the relationship between the step-size and the
accuracy of the estimate? |
|
| c) |
| i) | Central Difference |
| ii) | What is the relationship between the step-size and the
accuracy of the estimate? |
| iii) | By how much is the error reduced when the step-size is halved ? |
|
| 4. | |
Which method gives the best estimate?
| A. Forward difference approximation |
| B. Backward difference approximation |
| C. Central difference approximation |
|