Task 1
YOU MUST ANSWER BOTH PARTS (a) and (b)
Bob and Carol are researching electrical pulse waveforms used in neural stimulation of brains of Alzheimer patients. Bob says that research shows that optimal stimulation can be achieved using a waveform modeled by the function
i = 0.017e0.24t
where i is the current (amps) and t is pulse time in seconds.
(a) Tabulate values for the function over a suitable short time period, then illustrate the shape of the mathematical curve using techniques with which you are familiar.
(b) Carol says the current in the pulse ramps from 0.017 amps to a maximum of 0.187 amps, but wonders what the mean current might be over this period. Calculate an estimate of the mean current for her, using a technique with which you are familiar.
Opportunities for Higher Grading
M1 To achieve MI for task 1b you need to demonstrate a mathematical solution with which you are familiar, which produces an exact value. This will require the use of calculus over and above geometrical techniques.
D2 In part (a) you need to show independence of thought in how you illustrate the function including step by step planning and explanation demonstrating organized strategies.
Task 2 YOU MUST ANSWER ALL PARTS (a) to (d)
Ted and Alice are attempting to work out the flow rate of water in a river. The average velocity of the water is found to be 2.05m/sec according to a flow meter, however, they do not know the cross-section of the river. To find this, they decide to measure the depth of the river from one side of the embankment to the other using intervals of 1.5m. Below are their results. They propose to use a numerical method to determine the approximate cross-sectional area of the river and multiply this by the velocity (above) and therefore obtain the flow rate in m3/sec.
| Dist(m) |
0 |
1.5 |
3 |
4.5 |
6 |
7.5 |
9 |
10.5 |
12 |
13.5 |
15 |
| Depth(m) |
0 |
1.04 |
1.65 |
3.1 |
4.66 |
4.12 |
3.21 |
2.33 |
1.78 |
0.76 |
0 |
Ted says they can approximate the cross-sectional area of the river using the Trapezium rule. Alice suggests using Simpsons rule as she says it is more accurate.
(a) Using the data in the table, use the Trapezium rule to approximate the cross section of the river, and thereby determine the approximate flow rate to 2 d.p).
(b) Repeat your calculation, but this time use Simpsons rule.
(c) Compare your two answers.
Ted bell eves the cross-section of the river can be modelled by the mathematical function
y = 0.036x2 (where y is the depth (m) at distance x (m))
Alice doesn't think Ted is correct and suggests using the mid-ordinate rule as a rough check to see if it calculates similar values to those In (a) and (p) above.
(d) Using the table below to help you, use a further mathematical technique with 5 strips, each of width 3m, to work out the cross-sectional area of the river based on the model y=0.036x2. Does the answer compare favourably with the approximations obtained in (a) and (b), Explain your conclusion.
| Mid ordinate value (m) |
0.036x2 |
| 1.5 |
|
| 4.5 |
|
| 7.5 |
|
| 10.5 |
|
| 13.5 |
|
Task 3 YOU MUST ANSWER ALL PARTS (a) to (c)
Bob and arol are working on a problem in fluid mechanics. From experimentation they find that they can model the velocity flow, v (m/s), of a liquid along a channel using the following mathematical relationship
v3- 6v2 - 348v = -3112
Bob is to d that there should be a root of this equation between v = 10 and v = 11 m/s.
a) Re-arrange the equation and, using the interval above, find the root (to 3 decimal places) using the method of bisection.
(b) Carol knows a quicker way of doing this using the Newton-Raphson method. Using v=10 m/s as a first approximation, show how the Newton-Raphson method can be used to obtain the solution more quickly, and use it to check you answer in (a). Once again, obtain a solution to three decimal places.
(c) Use the Newton-Raphson procedure to find a root of the following equation to two decimal places, using r1 = -2 as a first approximation:
ex - 2x - 5 = 0