Let's break down the solution step by step: --- ### **a) Derive the Taylor series expansion for cos(x), expanding around \( x_0 = 0 \).** The Taylor series for a function \( f(x) \) about \( x_0 = 0 \) (Maclaurin series) is: \[ f(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 + \ldots \] For \( f(x) = \cos(x) \): - \( f(0) = \cos(0) = 1 \) - \( f'(x) = -\sin(x) \implies f'(0) = 0 \) - \( f''(x) = -\cos(x) \implies f''(0) = -1 \) - \( f'''(x) = \sin(x) \implies f'''(0) = 0 \) - \( f^{(4)}(x) = \cos(x) \implies f^{(4)}(0) = 1 \) So, the series is: \[ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \] Or, in general: \[ \boxed{ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} } \] --- ### **b) Use the Taylor series to solve for \( \cos(\pi/3) \) to 4 decimal places.** Given \( x = \pi/3 \approx 1.0472 \): \[ \cos\left(\frac{\pi}{3}\right) \approx 1 - \frac{(\pi/3)^2}{2!} + \frac{(\pi/3)^4}{4!} - \frac{(\pi/3)^6}{6!} + \cdots \] Let's compute the terms: - \( (\pi/3)^2 \approx 1.0966 \) - \( (\pi/3)^4 = (1.0966)^2 \approx 1.2026 \) - \( (\pi/3)^6 = (1.0966) \times 1.2026 \approx 1.3180 \) Now, the terms: - \( 1 \) - \( -\frac{1.0966}{2} = -0.5483 \) - \( +\frac{1.2026}{24} = +0.050108 \) - \( -\frac{1.3180}{720} = -0.00183 \) - Next term: \( +\frac{1.444}{40320} \approx +0.0000358 \) (very small) Add up: \[ 1 - 0.5483 + 0.050108 - 0.00183 + 0.0000358 \approx 0.5000 \] So, to 4 decimal places: \[ \boxed{ \cos(\pi/3) \approx 0.5000 } \] --- ### **c) Matlab Script** Here’s a sample Matlab script: ```matlab % Taylor series for cos(x), truncating after 100 terms angle_deg = input('Enter angle in degrees: '); x = deg2rad(angle_deg); % Convert to radians cos_taylor = 0; for n = 0:99 term = ((-1)^n) * (x^(2*n)) / factorial(2*n); cos_taylor = cos_taylor + term; end fprintf('cos(%d deg) ≈ %.10f\n', angle_deg, cos_taylor); % Verification fprintf('Built-in cos(%d deg) = %.10f\n', angle_deg, cos(x)); ``` **Verification:** Compare the output of the script to Matlab's built-in `cos(x)` function. --- ### **d) Approaches for Truncating the Series** **Different approaches:** 1. **Fixed number of terms:** - *Pros:* Simple to implement. - *Cons:* May be inefficient (too many terms) or inaccurate (too few terms) for some values of \( x \). 2. **Terminate when the next term is less than a tolerance (\( |a_n| < \epsilon \)):** - *Pros:* Ensures desired accuracy. - *Cons:* May require more computation to check each term. 3. **Terminate when the sum stops changing (converges):** - *Pros:* Avoids unnecessary calculations. - *Cons:* Can be affected by round-off error or oscillations. --- ### **e) Plots for Angles (0°, 100°, 200°, 300°, 400°) After 20 Terms** **Instructions:** - For each angle, plot the partial sum after each term (from 1 to 20). - Observe that for small angles the series converges quickly; for larger angles, convergence is slower. **Discussion:** - The further \( x \) is from 0, the more terms are needed for good accuracy. - For multiples of 360°, the value should return to 1. --- ### **f) Discussion on Round-off Error** **When/Where:** - For large \( n \) (many terms), terms become very small and can be swamped by round-off error. - For large \( x \), \( x^{2n} \) grows rapidly, so subtracting large numbers can cause loss of significance (catastrophic cancellation). **Strategies to Decrease Error:** - **Avoid large \( x \):** Use trigonometric identities to reduce \( x \) to a smaller equivalent. - **Increase precision:** Use higher-precision data types if available. - **Terminate earlier:** Stop adding terms once terms are negligible. - **Rearrange computations:** Group terms or use other formulations to minimize subtraction of nearly equal numbers. --- If you need code or plots for part (e), let me know!

Certainly! Let's go through each part step-by-step with detailed calculations and explanations. --- ## **a) Derive the Taylor series expansion for \( \cos(x) \) around \( x_0=0 \)** **Step 1:** Recall the Taylor series expansion about \( x_0=0 \): \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n} \] where \( f^{(n)}(0) \) is the \( n \)-th derivative evaluated at 0. **Step 2:** Find derivatives of \( \cos(x) \): | Derivative Order | Expression | Value at 0 | |------------------|----------------------------|-----------------------------------| | 0 | \( \cos(x) \) | \( \cos(0) = 1 \) | | 1 | \( -\sin(x) \) | \( -\sin(0) = 0 \) | | 2 | \( -\cos(x) \) | \( -\cos(0) = -1 \) | | 3 | \( \sin(x) \) | \( \sin(0) = 0 \) | | 4 | \( \cos(x) \) | \( \cos(0) = 1 \) | The derivatives repeat every 4 steps. **Step 3:** Write the series: \[ \cos(x) = \frac{\cos^{(0)}(0)}{0!} x^{0} + \frac{\cos^{(1)}(0)}{1!} x^{1} + \frac{\cos^{(2)}(0)}{2!} x^{2} + \frac{\cos^{(3)}(0)}{3!} x^{3} + \ldots \] Plugging in derivatives: \[ \cos(x) = 1 + 0 - \frac{x^2}{2!} + 0 + \frac{x^4}{4!} + 0 - \frac{x^6}{6!} + 0 + \ldots \] This simplifies to: \[ \boxed{ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} } \] --- ## **b) Approximate \( \cos(1/3) \) to 4 decimal places using the series** Given: - \( x = \frac{\pi}{3} \) **Step 1:** Calculate \( x \) numerically: \[ \pi \approx 3.1415926536 \Rightarrow x = \frac{\pi}{3} \approx 1.047197551 \approx 1.0472 \] **Step 2:** Compute each term: \[ \cos(x) \approx \sum_{n=0}^N \frac{(-1)^n x^{2n}}{(2n)!} \] Let's compute the first few terms until the sum stabilizes to 4 decimal places. --- ### Term 0 (\( n=0 \)): \[ a_0 = \frac{(-1)^0 x^{0}}{0!} = 1 \] --- ### Term 1 (\( n=1 \)): \[ a_1 = \frac{(-1)^1 x^{2}}{2!} = - \frac{x^2}{2} \] Calculate \( x^2 \): \[ x^2 = (1.0472)^2 \approx 1.0966 \] Then: \[ a_1 = - \frac{1.0966}{2} = -0.5483 \] Sum after 2 terms: \[ S_2 = 1 - 0.5483 = 0.4517 \] --- ### Term 2 (\( n=2 \)): \[ a_2 = \frac{(-1)^2 x^{4}}{4!} = \frac{x^4}{24} \] Calculate \( x^4 \): \[ x^4 = (x^2)^2 = (1.0966)^2 \approx 1.2026 \] Then: \[ a_2 = \frac{1.2026}{24} \approx 0.0501 \] Sum: \[ S_3 = 0.4517 + 0.0501 = 0.5018 \] --- ### Term 3 (\( n=3 \)): \[ a_3 = - \frac{x^6}{6!} \] Calculate \( x^6 = x^2 \times x^4 \): \[ x^6 = 1.0966 \times 1.2026 \approx 1.3180 \] Now, \[ a_3 = - \frac{1.3180}{720} \approx -0.00183 \] Sum: \[ S_4 = 0.5018 - 0.00183 = 0.49997 \approx 0.5000 \] --- ### Term 4 (\( n=4 \)): Calculate \( x^8 \): \[ x^8 = (x^4)^2 = (1.2026)^2 \approx 1.445 \] \[ a_4 = \frac{x^8}{8!} = \frac{1.445}{40320} \approx 0.0000358 \] Sum: \[ S_5 \approx 0.49997 + 0.000036 \approx 0.500006 \] **Observation:** The partial sums are approaching 0.5, as expected for \( \cos(\pi/3) \). **Final approximation:** \[ \boxed{ \cos(\pi/3) \approx 0.5000 } \] which matches the exact value \( \cos(\pi/3) = 0.5 \) to 4 decimal places. --- ## **Summary:** - The Taylor series for \( \cos(x) \): \[ \boxed{ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} } \] - Approximate \( \cos(\pi/3) \) using the first 4 terms yields approximately **0.5**, with high accuracy. --- If you'd like, I can also help with parts (c) to (f).

Certainly! Let's continue with parts (c) to (f), providing detailed explanations, calculations, and reasoning. --- ## **c) MATLAB Script to Compute Cosine Using Taylor Series (First 100 Terms)** ### **Objective:** Create a MATLAB script that: - Prompts the user to enter an angle in degrees. - Converts it to radians. - Calculates \( \cos(x) \) using the Taylor series truncated after 100 terms. - Displays the result. - Compares with MATLAB's built-in `cos()` function for verification. ### **MATLAB Script:** ```matlab % Prompt user for input angle_deg = input('Enter an angle in degrees: '); % Convert degrees to radians x = deg2rad(angle_deg); % Initialize sum cos_taylor = 0; % Loop through first 100 terms for n = 0:99 term = ((-1)^n) * (x^(2*n)) / factorial(2*n); cos_taylor = cos_taylor + term; end % Display results fprintf('Taylor series approximation (first 100 terms): %.10f\n', cos_taylor); fprintf('MATLAB built-in cos(%d degrees): %.10f\n', angle_deg, cos(x)); ``` ### **Verification:** - Run the script with various angles. - Observe that the Taylor series approximation converges to the built-in `cos()` value as the number of terms increases. - For most angles, 100 terms provide high accuracy. --- ## **d) Approaches for Terminating (Truncating) the Series** ### **1. Fixed Number of Terms** - **Method:** Stop after a predefined number of terms (e.g., 20, 50, 100). - **Pros:** Simple implementation; predictable computational effort. - **Cons:** May be inefficient for some \( x \) (too many terms) or insufficient (not enough for desired accuracy). ### **2. Threshold on Magnitude of Next Term** - **Method:** Continue adding terms until the absolute value of the next term is less than a specified tolerance \( \epsilon \). - **Pros:** Achieves desired precision; efficient. - **Cons:** Slight overhead in checking each term; possible issues if terms oscillate. ### **3. Convergence Criterion (Change in Sum)** - **Method:** Stop when the change in the sum between iterations is below a threshold. - **Pros:** Ensures convergence. - **Cons:** Might be affected by round-off errors; may require additional safeguards. --- ## **e) Plotting Series Partial Sums for Different Angles** ### **Objective:** - For angles \( 0^\circ, 100^\circ, 200^\circ, 300^\circ, 400^\circ \), - Compute the partial sum of the series after each of the first 20 terms. - Plot these partial sums vs. number of terms. - Discuss convergence behavior. ### **Approach:** ```matlab angles_deg = [0, 100, 200, 300, 400]; for angle_idx = 1:length(angles_deg) angle_deg = angles_deg(angle_idx); x = deg2rad(angle_deg); partial_sums = zeros(1,20); sum_series = 0; for n = 0:19 term = ((-1)^n) * (x^(2*n)) / factorial(2*n); sum_series = sum_series + term; partial_sums(n+1) = sum_series; end figure; plot(1:20, partial_sums, '-o'); title(sprintf('Partial Sums of cos(%d°) Series', angle_deg)); xlabel('Number of Terms'); ylabel('Partial Sum'); grid on; end ``` ### **Discussion:** - For small angles (e.g., 0°), convergence is very rapid. - For larger angles (e.g., 200°, 300°, 400°), convergence may be slower. - The series converges more quickly for angles near 0°, and more slowly for larger angles. --- ## **f) Round-Off Error Analysis and Strategies** ### **When and Where Round-Off Errors Are Most Likely:** - **Large number of terms:** when summing many small terms, subtractive cancellation can cause loss of precision. - **Large \( x \):** since \( x^{2n} \) grows rapidly, subtracting large numbers can lead to numerical instability. - **Near points where the series converges slowly:** more terms may accumulate errors. ### **Strategies to Reduce Round-Off Error:** 1. **Use Symmetry and Identities:** - Reduce \( x \) to a smaller equivalent angle using identities: \[ \cos(x) = \cos(x \pm 2\pi k) \] for some integer \( k \), to keep \( x \) small. 2. **Summation Order:** - Sum terms from smallest to largest magnitude to minimize rounding errors. 3. **Higher Precision Data Types:** - Use MATLAB's `vpa` (Variable Precision Arithmetic) or other high-precision libraries. 4. **Series Acceleration Techniques:** - Use Euler or Levin transformations to improve convergence. 5. **Early Termination:** - Stop adding terms once the incremental change is below a certain threshold, preventing unnecessary accumulation of errors. --- # **Summary:** - **Taylor series for \( \cos(x) \):** \[ \boxed{ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} } \] - **Approximation of \( \cos(\pi/3) \):** Using the first 4 terms, the series approximates \( 0.5 \) accurately. - **MATLAB implementation:** Allows user input, computes series sum, and compares to built-in function. - **Termination approaches:** Fixed terms, magnitude threshold, convergence criterion; each has pros and cons. - **Series convergence:** Faster for small angles; slower for large angles; visualized via plots. - **Round-off errors:** Most significant for many small terms and large \( x \); mitigated by reduction of \( x \), higher precision, and careful summation order. --- If you'd like further details, plots, or code outputs, feel free to ask!

Certainly! Here are the concise final answers for each part: --- ### **a) Derivation of the Taylor series for \( \cos(x) \) around \( x_0=0 \):** \[ \boxed{ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} } \] --- ### **b) Approximate \( \cos(1/3) \) to 4 decimal places:** Using the first four terms: \[ \cos(\pi/3) \approx 1 - \frac{(\pi/3)^2}{2} + \frac{(\pi/3)^4}{24} - \frac{(\pi/3)^6}{720} \approx 0.5000 \] **Final answer:** \(\boxed{0.5000}\) --- ### **c) MATLAB code snippet:** ```matlab angle_deg = input('Enter angle in degrees: '); x = deg2rad(angle_deg); cos_taylor = 0; for n=0:99 term = ((-1)^n)*(x^(2*n))/factorial(2*n); cos_taylor = cos_taylor + term; end fprintf('Approximate cos(%d°): %.10f\n', angle_deg, cos_taylor); fprintf('Built-in cos(%d°): %.10f\n', angle_deg, cos(x)); ``` *(Outputs will match closely, verifying the series approximation)* --- ### **d) Approaches for truncating the series:** 1. Fixed number of terms (e.g., 20, 100): Simple, predictable but potentially inefficient. 2. Term magnitude threshold (\( |a_n| < \epsilon \)): Efficient and accurate but requires checking each term. 3. Convergence criterion (change in sum): Ensures accuracy but may be affected by numerical noise. --- ### **e) Results of partial sum plots (truncating after 20 terms):** - For angles \( 0°, 100°, 200°, 300°, 400° \), the partial sums rapidly approach the true value for small angles. - Convergence is slower for larger angles, especially near \( 180° \) and beyond. - The plots show the number of terms needed for good accuracy varies with angle. --- ### **f) When and where round-off errors occur; strategies:** - **Most likely:** When summing many tiny terms for large \( n \) or large \( x \), causing loss of significance. - **Strategies:** - Reduce \( x \) using identities. - Sum terms from smallest to largest. - Use higher-precision arithmetic. - Terminate the series early when additional terms are negligible. --- Let me know if you'd like further elaboration!

Certainly! Here are the final concise answers for each part: **a)** \[ \boxed{ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} } \] **b)** Approximate \(\cos(\pi/3) \approx 0.5000\) (using four terms). **c)** MATLAB code: ```matlab angle_deg = input('Enter angle in degrees: '); x = deg2rad(angle_deg); cos_taylor = 0; for n=0:99 term = ((-1)^n)*(x^(2*n))/factorial(2*n); cos_taylor = cos_taylor + term; end fprintf('cos(%d°) ≈ %.10f\n', angle_deg, cos_taylor); fprintf('Built-in cos(%d°) = %.10f\n', angle_deg, cos(x)); ``` **d)** - Fixed number of terms: simple, but may be inaccurate. - Term magnitude threshold: efficient, ensures desired accuracy. - Convergence criterion: reliable but sensitive to numerical errors. **e)** Partial sums after 20 terms for angles 0°, 100°, 200°, 300°, 400° show rapid convergence for small angles; slower for larger angles. **f)** Round-off errors occur with many small terms and large \( x \); strategies include reducing \( x \) via identities, summing from smallest to largest, using higher precision, or early termination when terms are negligible.