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Numerical Methods for Engineers and Scientists: An Introduction With Applications Using MATLAB

73 SOLVED PROBLEMS

Question: 9.2

Evaluation of a single definite integral using fourth-order Gauss quadrature. Evaluate ∫0^3 e^-x² dx using four-point Gauss quadrature. ...

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Step 1: Since the limits of integration are [0,3],...
Question: 9.1

Distance traveled by a decelerating airplane. A Boeing 737-200 airplane of mass m = 97000 kg lands at a speed of 93 m/s (about 181 knots) and applies its thrust reversers at t = 0. The force F that is applied to the airplane, as it decelerates, is given by F = -5v² -570000, where vis the airplane’s ...

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Even though the governing equation is an ODE, it c...
Question: 8.7

Numerical partial differentiation. The following two-dimensional data for the x component of velocity u as a function of the two coordinates x and y is measured from an experiment: ...

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(a) In this part x_{i}=2, y_{i}=2, x_{i-1}=...
Question: 8.6

Comparing numerical and analytical differentiation. Consider the function f(x) = e^x. Write an expression for the first derivative of the function at x = 0 using the two-point central difference formula in Eq. (8.20). Investigate the effect that the spacing, h, between the points ...

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The two-point central difference formula in Eq. (8...
Question: 8.5

Using Richardson’s extrapolation in differentiation. Use Richardson’s extrapolation with the results in Example 8-4 to calculate a more accurate approximation for the derivative of the function f(x) = 2^x/x at the point x = 2 . Compare the results with the exact (analytical) derivative. ...

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In Example 8-4 two approximations of the derivativ...
Question: 8.4

Comparing numerical and analytical differentiation. Consider the function f(x) =2^x/x. Calculate the second derivative at x = 2 numerically with the three-point central difference formula using: (a) Points x = 1.8 , x = 2 , and x = 2.2 . (b) Points x=l.9, x=2,and x=2.1. Compare the results with the ...

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Analytical differentiation: The second derivative ...
Question: 8.3

Comparing numerical and analytical differentiation. Consider the function f(x) = x³. Calculate the first derivative at point x = 3 numerically with the three-point forward difference formula, using: (a) Points x = 3, x = 4, and x = 5. (b) Points x=3, x=3.25 ,and x=3.5 . Compare the results with the ...

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Analytical differentiation: The derivative of the ...
Question: 8.2

Damped vibrations. In a vibration experiment, a block of mass m is attached to a spring of stiffness k, and a dashpot with damping coefficient c, as shown in the figure. To start the experiment the block is moved from the equilibrium position and then released from rest. The position of the block ...

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(a) The velocity is calculated by using Eq. (8.7):...
Question: 8.1

Comparing numerical and analytical differentiation. Consider the function f(x) = x³. Calculate its first derivative at point x = 3 numerically with the forward, backward, and central finite difference formulas and using: (a) Points x = 2, x = 3, and x = 4. (b) Points x = 2.75, x = 3 , and x = 3.25. ...

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Analytical differentiation: The derivative of the ...
Question: 6.2

Curve fitting with a nonlinear function by writing the equation in a linear form. An experiment with an RC circuit is used for determining the capacitance of an unknown capacitor. In the circuit, shown on the right and in Fig. 6-8, a 5-MΩ resistor is connected in series to the unknown capacitor C ...

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It was shown in Fig. 6-9 that, as expected, an exp...
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