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free/or 0.5$ |
| m100739 | Generate eight equally-spaced points from the function
f(t) = sin2t
From t = 0 to 2(. Fit these data with
(a) A seventh-order interpolating polynomial and
(b) A cubic spline? |
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| m100741 | Given a square matrix [A], write a single line MATLAB command that will create a new matrix [Aug] that consists of the original matrix [A] augmented by an identity matrix [I]? |
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| m100742 | Given
f (x) = -2x6 - 1.5x 4 + 10x + 2
Use bisection to determine the maximum of this function. Employ initial guesses of xl = 0 and xu = 1, and perform iterations until the approximate relative error falls below 5%. |
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| m100745 | Given the initial conditions, y(0) = 1 and y (0) = 0, solve the following initial-value problem from t = 0 to 4:
d2y/dt2 + 4y = 0
Obtain your solutions with
(a) Euler s method and
(b) The fourthorder RK method.
In both cases, use a step size of 0.125. Plot both solutions on the same graph along with the exact solution y = cos 2t. |
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| m100747 | Given these data
Determine
(a) The mean,
(b) The standard deviation,
(c) The variance,
(d) The coefficient of variation, and
(e) The 95% confidence interval for the mean.
(f) Construct a histogram using a range from 7.5 to 11.5 with intervals of 0.5. |
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| m100748 | Given these data
Determine
(a) The mean,
(b) The standard deviation,
(c) The variance,
(d) The coefficient of variation, and
(e) The 90% confidence interval for the mean.
(f) Construct a histogram. Use a range from 28 to 34 with increments of 0.4.
(g) Assuming that the distribution is normal and that your estimate of the standard deviation is valid, compute the range (that is, the lower and the upper values) that encompasses 68% of the readings. Determine whether this is a valid estimate for the data in this problem. |
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| m100811 | In a similar fashion to the case study described in Sec. 16.4, develop the potential energy function for the system depicted in Fig. P16.32. Develop contour and surface plots in MATLAB. Minimize the potential energy function in order to determine the equilibrium displacements x1 and x2 given the forcing function F = 100 N, and the parameter ka = 20 and kb = 15 N/m?
Figure P16.32
Two frictionless masses connected to a wall by a pair of linear elastic springs. |
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| m100899 | Integration provides a means to compute how much mass enters or leaves a reactor over a specified time period, as in
Where t1 and t2 = the initial and final times, respectively. This formula makes intuitive sense if you recall the analogy between integration and summation. Thus, the integral represents the summation of the product of flow times concentration to give the total mass entering or leaving from t1 to t2. If the flow rate is constant, Q can be moved outside the integral:
Use numerical integration to evaluate this equation for the data listed below. Note that Q = 4 m3/min. |
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| m100911 | Just as Fourier s law and the heat balance can be employed to characterize temperature distribution, analogous relationships are available to model field problems in other areas of engineering. For example, electrical engineers use a similar approach when modeling electrostatic fields. Under a number of simplifying assumptions, an analog of Fourier s law can be represented in one-dimensional form as
Where D is called the electric flux density vector, e = permittivity of the material, and V = electrostatic potential. Similarly, a Poisson equation for electrostatic fields can be represented in one dimension as
Where ry = charge density. Use the finite-difference technique with ∆x = 2 to determine V for a wire where V(0) = 1000, V(20) = 0, e = 2, L = 20, and ry = 30. |
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| m101012 | Manning s equation can be used to compute the velocity of water in a rectangular open channel,
where U = velocity (m/s), S = channel slope, n = roughness coefficient, B = width (m), and H = depth (m). The following data are available for five channels:
Write a well-structured program that computes the velocity for each of these channels. Have the program display the input data along with the computed velocity in tabular form where velocity is the fifth column. Include headings on the table to label the columns. |
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| m101040 | Perform Crout decomposition on
2x1 - 5x2 + x3 = 12
-x1 + 3x2 - x3 = -8
3x1 - 4x2 + 2x3 = 16
Then, multiply the resulting [L] and [U] matrices to determine that [A] is produced? |
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| m101041 | Piecewise functions are sometimes useful when the relationship between a dependent and an independent variable cannot be adequately represented by a single equation. For example, the velocity of a rocket might be described by
Develop a well-structured function to compute v as a function of t. Then use this function to generate a table of v versus t for t 5 25 to 50 at increments of 0.5. |
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| m101047 | Prove that Eq. (22.15) is equivalent to Boole s rule.
Eq . 22.15 |
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| m101057 | Recall from Sec. 8.2 that determining the chemistry of water exposed to atmospheric CO2 can be determined by solving five nonlinear equations (Eqs. 8.6 through 8.10) for five unknowns: cT, [HCO-3], [CO32-], [H+], and [OH-]. Employing the parameters from Sec. 8.2 and the program developed in Prob. 9.20, solve this system for conditions in 1958 when the partial pressure of CO2 was 315 ppm. Use your results to compute the pH. |
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| m101058 | Recent interest in competitive and recreational cycling has meant that engineers have directed their skills toward the design
Figure P16.32
Two frictionless masses connected to a wall by a pair of linear elastic springs.
Figure P16.33
(a) A mountain bike along with (b) a free-body diagram for a part of the frame.
And testing of mountain bikes (Fig. P16.33a). Suppose that you are given the task of predicting the horizontal and vertical displacement of a bike bracketing system in response to a force. Assume the forces you must analyze can be simplified as depicted in Fig. P16.33b. You are interested in testing the response of the truss to a force exerted in any number of directions designated by the angle u. The parameters for the problem are E = Young s modulus 2 × 1011 Pa, A = cross-sectional area = 0.0001 m2, w = width = 0.44 m, l = length 5 0.56 m, and h 5 height 5 0.5 m. The displacements x and y can be solved by determining the values that yield a minimum potential energy. Determine the displacements for a force of 10,000 N and a range of u s from 08 (horizontal) to 908 (vertical). |
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| m101089 | Repeat Example 27.3, but insulate the left end of the rod. That is, change the boundary condition at the left end of the rod to T (0) = 0. |
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| m101100 | Repeat Prob. 20.41 but determine the coefficients of the polynomial (Sec. 18.4) that fit the data in Table P20.41.
Problem 20.41 |
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| m101125 | Solve the following initial value problem over the interval from t = 0 to 2 where y(0) = 1. Display all your results on the same graph.
dy /dt = yt2 - 1.1y
(a) Analytically.
(b) Euler s method with h = 0.5 and 0.25.
(c) Midpoint method with h = 0.5.
(d) Fourth-order RK method with h = 0.5. |
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| m101126 | Solve the following problem over the interval from x = 0 to 1 using a step size of 0.25 where y(0) = 1. Display all your results on the same graph.
dy/dt = (1 + 4t) √y
(a) Analytically.
(b) Euler s method.
(c) Heun s method without iteration.
(d) Ralston s method.
(e) Fourth-order RK method. |
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| m101144 | Suppose that a parachutist with linear drag (m = 70 kg, c = 12.5 kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 180 m/s relative to the ground.
(a) Write a system of four differential equations for x, y, ν x = dx/dt and νy = dy/dt.
(b) If the initial horizontal position is defined as x = 0, use Euler s methods with Δt = 1 s to compute the jumper s position over the first 10 s.
(c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open. |
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