Let's solve each part as requested, step-by-step, **with detailed calculations for the analytical derivative, numerical derivative, and percentage error** for each item, focusing first on **1.b and 2.b** as indicated in the table to fill in. --- ## **1. Compute the first derivative of:** ### **b. \( y = e^{x^2} \cos(x) + x^4 \sin(2x) \), at \( x=2 \), \( h=0.25 \)** ### **Step 1: Analytical Solution** #### **Find \( y'(x) \):** Let’s differentiate each term separately. 1. \( \frac{d}{dx}(e^{x^2} \cos(x)) \) - Use product rule: \( u = e^{x^2}, v = \cos(x) \) - \( u' = 2x e^{x^2} \), \( v' = -\sin(x) \) - So, \[ \frac{d}{dx}(e^{x^2} \cos(x)) = e^{x^2}(-\sin(x)) + 2x e^{x^2}\cos(x) \] 2. \( \frac{d}{dx}(x^4 \sin(2x)) \) - Product rule: \( u = x^4, v = \sin(2x) \) - \( u' = 4x^3, v' = 2\cos(2x) \) - So, \[ \frac{d}{dx}(x^4 \sin(2x)) = x^4 \cdot 2\cos(2x) + 4x^3 \sin(2x) \] Combine: \[ y'(x) = e^{x^2}(-\sin(x)) + 2x e^{x^2}\cos(x) + x^4 \cdot 2\cos(2x) + 4x^3 \sin(2x) \] #### **Evaluate at \( x=2 \):** - \( e^{2^2} = e^4 \approx 54.598 \) - \( \sin(2) \approx 0.9093 \) - \( \cos(2) \approx -0.4161 \) - \( \sin(4) \approx -0.7568 \) - \( \cos(4) \approx -0.6536 \) Now plug in \( x = 2 \): \[ \begin{align*} y'(2) = &\ e^4(-\sin(2)) + 2 \times 2 \times e^4 \cos(2) \\ &+ 2^4 \cdot 2 \cos(4) + 4 \times 2^3 \sin(4) \\ = &\ 54.598 \times (-0.9093) + 4 \times 54.598 \times (-0.4161) \\ &+ 16 \times 2 \times (-0.6536) + 4 \times 8 \times (-0.7568) \\ = &\ -49.614 + 54.598 \times 4 \times (-0.4161) + 32 \times (-0.6536) + 32 \times (-0.7568) \\ = &\ -49.614 + (54.598 \times 4 = 218.392; 218.392 \times -0.4161 = -90.850) \\ &+ (32 \times -0.6536 = -20.915) + (32 \times -0.7568 = -24.218) \\ \end{align*} \] Sum: \[ y'(2) = -49.614 - 90.850 - 20.915 - 24.218 = -185.597 \] **Analytical Solution:** \[ \boxed{-185.60} \] --- ### **Step 2: Numerical Solution (Central Difference)** **Central difference formula:** \[ f'(x) \approx \frac{f(x+h) - f(x-h)}{2h} \] Given \( h = 0.25 \), \( x = 2 \). Calculate \( f(2.25) \) and \( f(1.75) \): #### **Calculate \( f(2.25) \):** - \( e^{(2.25)^2} = e^{5.0625} \approx 157.88 \) - \( \cos(2.25) \approx -0.6297 \) - \( (2.25)^4 = 25.628 \) - \( \sin(4.5) \approx -0.9775 \) \[ f(2.25) = 157.88 \times -0.6297 + 25.628 \times -0.9775 = -99.404 + (-25.062) = -124.466 \] #### **Calculate \( f(1.75) \):** - \( e^{(1.75)^2} = e^{3.0625} \approx 21.39 \) - \( \cos(1.75) \approx -0.1782 \) - \( (1.75)^4 = 9.378 \) - \( \sin(3.5) \approx -0.3508 \) \[ f(1.75) = 21.39 \times -0.1782 + 9.378 \times -0.3508 = -3.813 + (-3.290) = -7.103 \] #### **Numerical Derivative:** \[ f'(2) \approx \frac{-124.466 - (-7.103)}{2 \times 0.25} = \frac{-117.363}{0.5} = -234.726 \] --- ### **Step 3: Percent Error** \[ \% \text{Error} = \left| \frac{\text{Numerical} - \text{Analytical}}{\text{Analytical}} \right| \times 100\% \] \[ = \left| \frac{-234.726 - (-185.60)}{-185.60} \right| \times 100\% = \left| \frac{-49.126}{-185.60} \right| \times 100\% = 26.48\% \] --- ### **Fill Table Row 1b:** | Item | Analytical Solution | Numerical Solution | % Error | |------|--------------------|-------------------|---------| | 1b | -185.60 | -234.73 | 26.48% | --- ## **2. Compute the second derivative:** ### **b. \( y = e^{-2x} - 5x \), at \( x=0 \), \( h=0.25 \)** ### **Step 1: Analytical Solution** First derivative: \[ y'(x) = -2 e^{-2x} - 5 \] Second derivative: \[ y''(x) = 4 e^{-2x} \] At \( x = 0 \): \[ y''(0) = 4 \times e^0 = 4 \] --- ### **Step 2: Numerical Solution (Central Difference, Second Derivative)** \[ f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} \] At \( x=0 \), \( h = 0.25 \): Calculate \( f(0.25) \), \( f(0) \), \( f(-0.25) \): - \( f(0.25) = e^{-2 \times 0.25} - 5 \times 0.25 = e^{-0.5} - 1.25 \approx 0.6065 - 1.25 = -0.6435 \) - \( f(0) = e^{0} - 0 = 1 \) - \( f(-0.25) = e^{-2 \times -0.25} - 5 \times (-0.25) = e^{0.5} + 1.25 \approx 1.6487 + 1.25 = 2.8987 \) \[ f''(0) \approx \frac{-0.6435 - 2 \times 1 + 2.8987}{(0.25)^2} = \frac{-0.6435 - 2 + 2.8987}{0.0625} = \frac{0.2552}{0.0625} = 4.083 \] --- ### **Step 3: Percent Error** \[ \% \text{Error} = \left| \frac{4.083 - 4}{4} \right| \times 100\% = 2.075\% \] --- ### **Fill Table Row 2b:** | Item | Analytical Solution | Numerical Solution | % Error | |------|--------------------|-------------------|---------| | 2b | 4.00 | 4.08 | 2.08% | --- ## **Final Table (Partial Fill):** | Item | Analytical Solution | Numerical Solution | % Error | |------|--------------------|-------------------|---------| | 1b | -185.60 | -234.73 | 26.48% | | 2b | 4.00 | 4.08 | 2.08% | If you want the rest of the items (1a, 1c, 2a, 2c) solved, let me know!

Certainly! Below is the **comprehensive, step-by-step solution** for **all parts** of the question, including **analytical derivatives, numerical derivatives, percentage errors**, and **final answers**. --- ## **Part 1: First Derivatives** ### **1.a. \( y = 3x^2 - 12x + 4 \) at \( x=25 \), \( h=0.2 \)** #### **Analytical derivative:** \[ y' = \frac{d}{dx}(3x^2 - 12x + 4) = 6x - 12 \] At \( x=25 \): \[ y'(25) = 6 \times 25 - 12 = 150 - 12 = 138 \] #### **Numerical derivative (central difference):** \[ f'(x) \approx \frac{f(x+h) - f(x-h)}{2h} \] Calculate \( f(25 + 0.2) = f(25.2) \): \[ f(25.2) = 3(25.2)^2 - 12(25.2) + 4 \] \[ (25.2)^2 = 635.04 \] \[ f(25.2) = 3 \times 635.04 - 12 \times 25.2 + 4 = 1905.12 - 302.4 + 4 = 1606.72 \] Calculate \( f(24.8) \): \[ (24.8)^2 = 615.04 \] \[ f(24.8) = 3 \times 615.04 - 12 \times 24.8 + 4 = 1845.12 - 297.6 + 4 = 1551.52 \] Numerical derivative: \[ f'(25) \approx \frac{1606.72 - 1551.52}{2 \times 0.2} = \frac{55.2}{0.4} = 138 \] **Percent error:** \[ \left| \frac{138 - 138}{138} \right| \times 100\% = 0\% \] --- ### **1.b. \( y = e^{x^2} \cos x + x^4 \sin 2x \) at \( x=2 \), \( h=0.25 \)** *(Already computed above in detail)* - **Analytical derivative:** \[ y' = e^{x^2}(-\sin x) + 2x e^{x^2} \cos x + 2 x^4 \cos 2x + 4 x^3 \sin 2x \] At \( x=2 \), values: \[ \boxed{y' \approx -185.60} \] - **Numerical derivative:** \[ f'(2) \approx \frac{f(2.25) - f(1.75)}{2 \times 0.25} = \frac{-124.47 - (-7.10)}{0.5} = \frac{-117.37}{0.5} = -234.74 \] - **Percent error:** \[ \left| \frac{-234.74 - (-185.60)}{-185.60} \right| \times 100\% \approx 26.48\% \] --- ### **1.c. \( y = \ln x \) at \( x=1 \), \( h=0.2 \)** - **Analytical derivative:** \[ y' = \frac{1}{x} \Rightarrow y'(1) = 1 \] - **Numerical derivative:** \[ f'(1) \approx \frac{\ln(1 + 0.2) - \ln(1 - 0.2)}{2 \times 0.2} = \frac{\ln(1.2) - \ln(0.8)}{0.4} \] Calculate: \[ \ln(1.2) \approx 0.1823, \quad \ln(0.8) \approx -0.2231 \] \[ f'(1) \approx \frac{0.1823 - (-0.2231)}{0.4} = \frac{0.4054}{0.4} = 1.0135 \] - **Percent error:** \[ \left| \frac{1.0135 - 1}{1} \right| \times 100\% = 1.35\% \] --- ## **Part 2: Second Derivatives** ### **2.a. \( y = \tan \frac{x}{4} \), at \( x=3 \), \( h=0.5 \)** - **Analytical second derivative:** \[ y' = \frac{1}{4} \sec^2 \frac{x}{4} \] \[ y'' = \frac{d}{dx} \left( \frac{1}{4} \sec^2 \frac{x}{4} \right ) = \frac{1}{4} \times 2 \sec^2 \frac{x}{4} \tan \frac{x}{4} \times \frac{1}{4} = \frac{1}{8} \sec^2 \frac{x}{4} \tan \frac{x}{4} \] At \( x=3 \): \[ u = \frac{3}{4} = 0.75 \] \[ \sec^2 0.75 = \frac{1}{\cos^2 0.75} \approx \frac{1}{0.7317^2} \approx 1.872 \] \[ \tan 0.75 \approx 0.9316 \] \[ y''(3) \approx \frac{1}{8} \times 1.872 \times 0.9316 \approx 0.217 \] - **Numerical second derivative:** \[ f''(x) \approx \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} \] Calculate: \[ f(3.5) = \tan(3.5/4) = \tan(0.875) \approx 1.213 \] \[ f(2.5) = \tan(2.5/4) = \tan(0.625) \approx 0.726 \] \[ f(3) \text{ as above} \approx 0.9316 \] Second derivative: \[ \approx \frac{1.213 - 2 \times 0.9316 + 0.726}{(0.5)^2} = \frac{1.213 - 1.8632 + 0.726}{0.25} = \frac{0.0758}{0.25} = 0.303 \] - **Percent error:** \[ |0.303 - 0.217| / 0.217 \times 100\% \approx 39.6\% \] --- ### **2.b. \( y = e^{-2x} - 5x \), at \( x=0 \), \( h=0.25 \)** *(Already computed above)* - **Analytical second derivative:** \[ y'' = 4 e^{-2x} \] At \( x=0 \): \[ y''(0) = 4 \times 1 = 4 \] - **Numerical second derivative:** \[ f(0.25) = e^{-0.5} - 1.25 \approx 0.6065 - 1.25 = -0.6435 \] \[ f(0) = 1 \] \[ f(-0.25) = e^{0.5} + 1.25 \approx 1.6487 + 1.25 = 2.8987 \] \[ f''(0) \approx \frac{-0.6435 - 2 \times 1 + 2.8987}{0.0625} = \frac{0.2552}{0.0625} = 4.083 \] **Percent error:** \[ |4.083 - 4|/4 \times 100\% \approx 2.075\% \] --- ### **2.c. \( y = \sin x \) at \( x=2 \), \( h=0.125 \)** - **Analytical second derivative:** \[ y'' = -\sin x \] At \( x=2 \): \[ y'' = - \sin 2 \approx -0.9093 \] - **Numerical second derivative:** \[ f(2+h) = \sin(2 + 0.125) = \sin(2.125) \approx 0.828 \] \[ f(2) = \sin 2 \approx 0.9093 \] \[ f(2 - h) = \sin(1.875) \approx 0.954 \] \[ f''(2) \approx \frac{0.828 - 2 \times 0.9093 + 0.954}{(0.125)^2} = \frac{0.828 - 1.8186 + 0.954}{0.015625} = \frac{-0.0366}{0.015625} \approx -2.342 \] - **Percent error:** \[ | -2.342 - (-0.9093)|/0.9093 \times 100\% \approx 157.5\% \] --- ## **Summary Table of Results** | Part | Function | Analytical Derivative / Second Derivative | Numerical Approximation | % Error | |--------|--------------|-------------------------------------------|------------------------|---------| | 1.a | \( 3x^2 - 12x + 4 \) | 138 | 138 | 0% | | 1.b | \( e^{x^{2}} \cos x + x^{4} \sin 2x \) | -185.60 | -234.74 | 26.48% | | 1.c | \( \ln x \) | 1 | 1.0135 | 1.35% | | 2.a | \( \tan \frac{x}{4} \) | 0.217 | 0.303 | 39.6% | | 2.b | \( e^{-2x} - 5x \) | 4 | 4.083 | 2.08% | | 2.c | \( \sin x \) | -0.9093 | -2.342 | 157.5% | --- **Note:** The percentage error indicates how close the numerical approximation is to the analytical derivative/second derivative. Some functions with rapid change or high curvature can have larger errors. --- **If you need further clarification or additional parts, feel free to ask!**