Question

Use zero- through third-order Taylor series expansions to predict f(2, 7) for f(x) = cos(3x) using a base point of x_0 = 2.5. For each approximation, list the following:

a. Approximate value of f(2, 7)
b. On what order of the step size h is the truncation error?
c. Compute the true percent relative errors, epsilon_t, for each approximation.

Answer 1

Following are the solution to this question:

Explanation:

Given:

`\bold{f(x) = cos(3x)}\\\to f'(x) = - 3 sin (3x)\\\to f''(x) = - 9 cos (3x)\\\to f'''(x) = 27 sin (3x)\\`

Similarly:

`\bold{f(2.5)' = cos(6.75)}\\\to f'(2.5) = - 3 sin (6.75)\\\to f''(2.5) = - 9 cos (6.75)\\\to f'''(2.5) = 27 sin (2.5)\\`

calculating the order values
`\longrightarrow \cos(6.75)`:

In the 1st order:

`\to cos(6.75) + (x-2.5) 3 sin (6.75)`

In the 2nd order:

`\to cos(6.75) + [1- ((x-2.5)^2)/(2)9] + (x-2.5) 3 sin (6.75)`

In the 3rd order:

`\to cos(6.75) + [1- (9(x-2.5)^2)/(2)] + 3 sin (6.75)(3(x-2.5) - (27)/(6) (1-2.5))`

In point a:

`\boxed{\left\begin{array}{cccc}{Zero&First &Second&Third}\\0.993068457& 0.420690485&9.2635909&16.0135909\\\end{array}\right}`

In point b:

`\boxed{\left\begin{array}{cccc}{First &Second&Third&Fourth}\\0.993068457& 0.420690485&9.2635909&16.0135909\\\end{array}\right}`

In point C:

At this point, data is missing.