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Write a script to solve these Problems.
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1. Create a vector of the values of ex cos(x) at x = 3, 3.1, 3.2, . . . , 6.
2. Create the following vectors:
(a) (0.13 0.2 1, 0.1 60.2 4,…, 0.1 360.2 34)
(b) ( 2,
22 23
2
,
3
,… ,
225
25
)
1. Calculate the following:
3
2
(a) ∑100
𝑖=10(𝑖 + 4𝑖 )
2𝑖
3𝑖
𝑖
𝑖
(b) ∑25
𝑖=1 ( + 2 )
2. By using the function cumprod or otherwise, calculate
1+
2
24
246
2 4 38
) + ⋯+ ( … )
+( )+(
3
35
357
3 5 39
3. Solve the following system of linear equations in five unknowns
x1 + 2x2 + 3x3 + 4x4 + 5x5 = 7
2x1 + x2 + 2x3 + 3x4 + 4x5 = −1
3x1 + 2x2 + x3 + 2x4 + 3x5 = −3
4x1 + 3x2 + 2x3 + x4 + 2x5 = 5
5x1 + 4x2 + 3x3 + 2x4 + x5 = 17
By considering an appropriate matrix equation Ax = y.
Make use of the special form of the matrix A. The method used for the solution should
easily generalise to a larger set of equations where the matrix A has the same structure;
hence the solution should not involve typing in every number of A.
4. Write a function tmpFn(xVec) such that if xVec is the vector x = (x1, . . . , xn) then
tmpFn(xVec) returns the vector of moving averages:
𝑥1 + 𝑥2 + 𝑥3 𝑥2 + 𝑥3 + 𝑥4
(𝑥𝑛−2 + 𝑥𝑛−1 + 𝑥𝑛 )
,
,...,
3
3
3
Try out your function; for example, try tmpFn( c(1:5,6:1) ).
5. Suppose x0 = 1 and x1 = 2 and
𝑥𝑗 = 𝑥𝑗−1 +
2
𝑓𝑜𝑟 𝑗 = 1,2, …
𝑥 (𝑗 − 1)
Write a function testLoop which takes the single argument n and returns the first n −
1 values of the sequence {xj}j≥0: that means the values of x0, x1, x2, . . . , xn−2.