$h = \frac{1}{2}, h = \frac{1}{4}, h = \frac{1}{2}$ $T_o(\frac{1}{2}) = T_o(h) = \frac{h}{2} [f(x_o) + f(x_n)] + h \sum_{i=1}^{n-1} f(x_i)$ $= \frac{1}{2} \cdot \frac{1}{2} [\frac{2}{\frac{1}{2}} + \frac{2}{\frac{3}{2}}] + \frac{1}{2} \cdot \frac{2}{\frac{3}{2}} = \frac{1}{4} [6\frac{2}{3}] + \frac{1}{3} = \frac{3}{8} + \frac{1}{3} = \frac{17}{24} \approx 0.70833$ $T_o(\frac{1}{4}) = \frac{1}{2} (\frac{1}{4} + \frac{4}{7}) + \frac{1}{4} (\frac{4}{5} + \frac{5.7}{5.5} + \frac{5.6}{6.7})$ $= \frac{1}{8} (4 + \frac{4(2)}{210} + \frac{35}{210} + \frac{30}{210}) = \frac{1}{8} (4 + \frac{62}{210} + \frac{35}{210} + \frac{30}{210})$ $T_1 = \frac{4 \cdot T_o(\frac{1}{4}) - T_o(1)}{3}$ $= \frac{4 \cdot \frac{17}{24} - \frac{3}{8}}{3} = \frac{68 - 18}{24 \times 3} = \frac{50}{72} \approx 0.69444$ $f(x) = \frac{1}{x}$ $\frac{34 - 9}{36} = \frac{25}{36}$
Added by Natalia M.
Close
Step 1
Step 1: The problem is to approximate the integral $\int_1^2 \frac{1}{x} dx$ using trapezoidal rule with $h = \frac{1}{2}$ and $h = \frac{1}{4}$. Show more…
Show all steps
Your feedback will help us improve your experience
Adi S and 89 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Determined approximations, as accurately as possible, for each missing entry in the following table using Five-point formulas.
Adi S.
2. Use the Newton-Raphson method to approximate to two decimal places the root of f(x) = e^{-x} - x in the interval [0, 1] using x = 0 as the initial guess to start the iteration. 3. Use the secant method to approximate to two decimal places the root of f(x) = e^{-x} - x in the interval [0, 1] the endpoints of the interval as initial guesses to start the iteration. 4. Use the Lagrange interpolating polynomial to estimate the value of f(x) = 10xe^{-x} at x = 3 using the values of f(x) at x = 1, x = 2, and x = 4. 5. Estimate the value of the unknown function using Lagrange interpolation if the known values of the function are as follows: x_k 1.5 2.4 3.2 4.3 y_k 2.5 5.2 1.8 1.4 7. Calculate the first derivative of f(x) = 10xe^{-x} at x = 0 using the forward difference formula with step sizes of: h = 0.1, h = 0.01, and h = 0.001. 8. Calculate the first derivative of f(x) = 10xe^{-x} at x = 0 using the backward difference formula with step sizes of: h = 0.1, h = 0.01, and h = 0.001. 9. Calculate the first derivative of f(x) = 10xe^{-x} at x = 0 using the central difference formula with step sizes of: h = 0.1, h = 0.01, and h = 0.001. 10. Calculate the second derivative of f(x) = 10xe^{-x} at x = 0 using the central difference formula with step sizes of: h = 0.1, h = 0.01, and h = 0.001. 11. Integrate f(x) = 10xe^{-x} over the interval [0, 1] with n = 2, n = 4, and n = 8 sub-intervals using the trapezoidal rule. 12. Integrate f(x) = 10xe^{-x} over the interval [0, 1] with n = 2, n = 4, and n = 8 sub-intervals using the Simpson's rule.
Sri K.
Suppose the following data points are generated by a smooth function f(x): x: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2 f(x): 0.6931, 0.6951, 0.7239, 0.8400, 1.0986, 1.4910, 1.9548, 2.4318, 2.8904 Using Romberg integration, find an O(h^6) approximation of ∠₀ ² f(x) dx. ( ie compute R₃₃) 1.5112 0.8036 2.7552 3.3930
Recommended Textbooks
Elementary and Intermediate Algebra
Algebra and Trigonometry
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD