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Task 3.3.4

The document discusses the formulation and expression of functions using Taylor series, confirming the correctness of various series expansions. It highlights the trade-off between computational feasibility and exact solutions, emphasizing the use of finite truncation for practical approximations. Additionally, it provides algorithms and implementations in Python and Octave for approximating functions like e^x and cos(x), noting that higher terms improve accuracy while built-in functions are generally more efficient.

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12 views3 pages

Task 3.3.4

The document discusses the formulation and expression of functions using Taylor series, confirming the correctness of various series expansions. It highlights the trade-off between computational feasibility and exact solutions, emphasizing the use of finite truncation for practical approximations. Additionally, it provides algorithms and implementations in Python and Octave for approximating functions like e^x and cos(x), noting that higher terms improve accuracy while built-in functions are generally more efficient.

Uploaded by

fagbuleolakunle223
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Task 04: Formulating and Expressing Functions Using Polynomials

1. Confirm correctness of functions in Table 2.9


All listed series are correct Taylor Series expansions:
 e^x: (_{i=0}^)
 cos(x): (_{i=0}^)
 sin(x): (_{i=0}^)
 sinh(x), cosh(x), ln(1/(1 - x)), etc. are also correctly represented.
2. Computational Feasibility vs. Exact Solutions
Computability:
 Finite-term approximations yield practical numerical results
 Achievable through iterative summation of series terms

Limitations for Exact Solutions:


 Infinite terms required for perfect accuracy
 No general algebraic solution exists across all x-values
 Numerical precision constrained by floating-point representation3. Making them
solvable

3. Making them solvable

 Use finite truncation (Taylor approximation).


 Select an acceptable number of terms (n) for required precision.
 Apply error analysis using first omitted term.

4. Algorithm for Taylor Series (General)


Function taylor_series(fx, x, n):
Initialize sum = 0
For i from 0 to n:
Compute term = function-specific formula (e.g., x^i / i!)
Add term to sum
Return sum

5. Implementation in Python (Example: e^x)


from math import factorial, exp
import numpy as np

def taylor_exp(x, n):


return sum([(x**i)/factorial(i) for i in range(n)])
x_vals = np.arange(0, np.pi/2 + 0.1, 0.5)
n_terms = [5, 20, 30, 100, 500]

for n in n_terms:
print(f"n = {n}")
for x in x_vals:
approx = taylor_exp(x, n)
exact = exp(x)
print(f"x = {x:.2f}, Approx = {approx:.6f}, Exact =
{exact:.6f}, Error = {abs(approx - exact):.2e}")

6. Octave Implementation for cos(x) with Plot


function y = taylor_cos(x, n)
y = 0;
for i = 0:n
y += ((-1)^i * x^(2*i)) / factorial(2*i);
end
endfunction

x_vals = 0:0.5:pi/2;
orders = [5, 20, 30, 100, 500];

for n = orders
approx = arrayfun(@(x) taylor_cos(x, n), x_vals);
actual = cos(x_vals);
plot(x_vals, approx, 'DisplayName', ['n = ', num2str(n)]);
hold on;
end

plot(x_vals, cos(x_vals), 'k--', 'DisplayName', 'Built-in cos(x)');


legend();
title('Comparison of Taylor cos(x) approximations');
xlabel('x'); ylabel('cos(x)');
grid on;

7. Observations
 Higher n leads to better approximation.
 At n = 20 or above, results are almost indistinguishable from built-in values.
 Errors decrease significantly with more terms.

8. Comparison with Built-in Functions


 Built-in functions are optimized and usually faster and more precise.
 Taylor series is effective for education and manual approximation.
 Ideal for understanding the basis of function evaluation in computation.

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