The following table represents the prices of copper futures (in USD) from theperiod of time from 14.06.2020 to 06.06.2021, where t is time since 14.06.2020 (measured in weeks) and f(t) is the corresponding price: t f(t) 0 2.6655 1 2.7120 2 2.7800 3 2.9255 4 2.9355 5 2.9220 6 2.9110 7 2.8445 8 2.9070 t f(t) 9 2.9570 10 3.0365 11 3.0810 12 3.0590 13 3.1215 14 2.9785 15 2.9900 16 3.0800 17 3.0720 t f(t) 18 3.1305 19 3.0565 20 3.1635 21 3.1890 22 3.3125 23 3.4210 24 3.5285 25 3.5340 26 3.6370 t f(t) 27 3.5675 28 3.5240 29 3.6800 30 3.6070 31 3.6315 32 3.5565 33 3.6265 34 3.7885 35 4.0775 t f(t) 36 4.0925 37 4.0755 38 4.1400 39 4.1130 40 4.0680 41 3.9905 42 4.0400 43 4.1680 44 4.3360 t f(t) 45 4.4680 46 4.7485 47 4.6545 48 4.4810 49 4.6775 50 4.5290 51 4.5375 Let L(x) be the third Lagrange polynomial for the function f(x) with the nodes X0 = 0, x1 = 12, x2 = 42, x3 = 50. (i) Find the value L(22) of the Lagrange polynomial at x = 22 and the relative error in the approximation f(22) ≈ L(22): (ii) Find the value L(51) of the Lagrange polynomial at x = 51 and the relative error in the approximation f(51) ≈ L(51): All calculations are to be carried out in the FPA5, and the computational results are to be presented in two standard output tables for themethod of the form xk x0 x1 … xn yk= f(xk) y0 y1 … yn Lk(x) L0(x) L1(x) … Ln(x) Total, L(x) ykLk(x) y0L0(x) y1L1(x) … ynLn(x) ∑kykLk(x)
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
The following table represents the prices of copper futures (in USD) from theperiod of time from 14.06.2020 to 06.06.2021, where t is time since 14.06.2020 (measured in weeks) and f(t) is the corresponding price:
|
t f(t) |
0 2.6655 |
1 2.7120 |
2 2.7800 |
3 2.9255 |
4 2.9355 |
5 2.9220 |
6 2.9110 |
7 2.8445 |
8 2.9070 |
|
t f(t) |
9 2.9570 |
10 3.0365 |
11 3.0810 |
12 3.0590 |
13 3.1215 |
14 2.9785 |
15 2.9900 |
16 3.0800 |
17 3.0720 |
|
t f(t) |
18 3.1305 |
19 3.0565 |
20 3.1635 |
21 3.1890 |
22 3.3125 |
23 3.4210 |
24 3.5285 |
25 3.5340 |
26 3.6370 |
|
t f(t) |
27 3.5675 |
28 3.5240 |
29 3.6800 |
30 3.6070 |
31 3.6315 |
32 3.5565 |
33 3.6265 |
34 3.7885 |
35 4.0775 |
|
t f(t) |
36 4.0925 |
37 4.0755 |
38 4.1400 |
39 4.1130 |
40 4.0680 |
41 3.9905 |
42 4.0400 |
43 4.1680 |
44 4.3360 |
|
t f(t) |
45 4.4680 |
46 4.7485 |
47 4.6545 |
48 4.4810 |
49 4.6775 |
50 4.5290 |
51 4.5375 |
|
|
Let L(x) be the third Lagrange polynomial for the function f(x) with the nodes
X0 = 0, x1 = 12, x2 = 42, x3 = 50.
(i) Find the value L(22) of the Lagrange polynomial at x = 22 and the relative error in the approximation f(22) ≈ L(22):
(ii) Find the value L(51) of the Lagrange polynomial at x = 51 and the relative error in the approximation f(51) ≈ L(51):
All calculations are to be carried out in the FPA5, and the computational results are to be presented in two standard output tables for themethod of the form
|
xk |
x0 |
x1 |
… |
xn |
|
|
yk= f(xk) |
y0 |
y1 |
… |
yn |
|
|
Lk(x) |
L0(x) |
L1(x) |
… |
Ln(x) |
Total, L(x) |
|
ykLk(x) |
y0L0(x) |
y1L1(x) |
… |
ynLn(x) |
∑kykLk(x) |
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