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m52336  5. Use Algorithm 5.4 to approximate the solutions to the initialvalue problems in Exercise 1
In Exercise 1
a. y = te3t − 2y, 0≤ t ≤ 1, y(0) = 0, with h=0.2; actual solution y(t) = 1/5 te3t  1/25 e3t +1/25 e−2t .
b. y = 1 + (t − y)2, 2≤ t ≤ 3, y(2) = 1, with h = 0.2; actual solution y(t) = t + 1/1−t .
c. y = 1 + y/t, 1≤ t ≤ 2, y(1) = 2, with h = 0.2; actual solution y(t) = t ln t + 2t.
d. y = cos 2t + sin 3t, 0 ≤ t ≤ 1, y(0) = 1, with h = 0.2; actual solution y(t) = 1/2 sin 2t  1/3 cos 3t + 4/3 . 
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m52338  A 6cm by 5cm rectangular silver plate has heat being uniformly generated at each point at the rate q = 1.5 cal/cm3·s. Let x represent the distance along the edge of the plate of length 6 cm and y be the distance along the edge of the plate of length 5 cm. Suppose the temperature u along the edges is kept at the following temperatures:
u(x, 0) = x(6 − x), u(x, 5) = 0,....................0 ≤ x ≤ 6,
u(0, y) = y(5 − y), u(6, y) = 0, ....................0 ≤ y ≤ 5,
where the origin lies at a corner of the plate with coordinates (0, 0) and the edges lie along the positive x and yaxes. The steadystate temperature u = u(x, y) satisfies Poisson s equation:
∂2u / ∂x2 (x, y) + ∂2u / ∂y2 (x, y) = −q/K, 0< x < 6, 0 < y < 5,
where K, the thermal conductivity, is 1.04 cal/cm·deg·s. Approximate the temperature u(x, y) using Algorithm 12.1 with h = 0.4 and k = 1/3. 
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m52345  a. Analyze the roundoff errors, as in Example 4, for the formula
b. Find an optimal h > 0 for the function given in Example 2. 
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m52347  a. Approximate f (0.05) using the following data and the Newton forwarddifference formula:
b. Use the Newton backwarddifference formula to approximate f (0.65).
c. Use Stirling s formula to approximate f (0.43). 
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m52377  A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second
a. Use a Hermite polynomial to predict the position of the car and its speed when t = 10 s.
b. Use the derivative of the Hermite polynomial to determine whether the car ever exceeds a 55 mi/h speed limit on the road. If so, what is the first time the car exceeds this speed?
c. What is the predicted maximum speed for the car? 
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m52378  A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second.
a. Use a clamped cubic spline to predict the position of the car and its speed when t = 10 s.
b. Use the derivative of the spline to determine whether the car ever exceeds a 55mi/h speed limit on the road; if so, what is the first time the car exceeds this speed?
c. What is the predicted maximum speed for the car? 
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m52393  a. Change Algorithm 11.2 to incorporate the Secant method instead of Newton s method. Use t0 = (β − α) / (b − a) and t1 = t0 + (β  y (b, t0)) / (b − a).
b. Repeat Exercise 4(a) and 4(c) using the Secant algorithm derived in part (a), and compare the number of iterations required for the two methods.
In exercise 4(a) and 4(c) 
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m52401  A coaxial cable is made of a 0.1in.square inner conductor and a 0.5in.square outer conductor.
The potential at a point in the cross section of the cable is described by Laplace s equation. Suppose the inner conductor is kept at 0 volts and the outer conductor is kept at 110 volts. Find the potential between the two conductors by placing a grid with horizontal mesh spacing h = 0.1 in. and vertical mesh spacing k = 0.1 in. on the region
D = {(x, y)  0 ≤ x, y ≤ 0.5}. 
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m52446  a. Derive an estimate for E(f ) in the Composite Trapezoidal rule using the method in Exercise 16.
b. Repeat part (a) for the Composite Midpoint rule. 
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m52447  a. Derive the AdamsBashforth TwoStep method by using the Lagrange form of the interpolating polynomial.
b. Derive the AdamsBashforth FourStep method by using Newton s backwarddifference form of the interpolating polynomial. 
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m52449  a. Determine the discrete least squares trigonometric polynomial S4(x), using m = 16, for f (x) = x2 sin x on the interval [0, 1].
b. Compute
c. Compare the integral in part (b) to 
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m52450  a. Determine the trigonometric interpolating polynomial S4(x) of degree 4 for f (x) = x2 sin x on the interval [0, 1].
b. Compute
c. Compare the integral in part (b) to 
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m52454  A drug administered to a patient produces a concentration in the blood stream given by c(t) = Ate−t/3 milligrams per milliliter, t hours after A units have been injected. The maximum safe concentration is 1 mg/mL.
a. What amount should be injected to reach this maximum safe concentration, and when does this maximum occur?
b. An additional amount of this drug is to be administered to the patient after the concentration falls to 0.25 mg/mL. Determine, to the nearest minute, when this second injection should be given.
c. Assume that the concentration from consecutive injections is additive and that 75% of the amount originally injected is administered in the second injection. When is it time for the third injection? 
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m52513  A Fredholm integral equation of the second kind is an equation of the form
Where a and b and the functions f and K are given. To approximate the function u on the interval [a, b], a partition x0 = a < x1 < · · · < xm−1 < xm = b is selected and the equations
Are solved for u(x0), u(x1), · · · , u(xm). The integrals are approximated using quadrature formulas based on the nodes x0, · ·, xm. In our problem, a = 0, b = 1, f (x) = x2, and K(x, t) = ex−t.
a. Show that the linear system
u(0) = f (0) + 1/2 [K(0, 0)u(0) + K(0, 1)u(1)],
u(1) = f (1) + 1/2 [K(1, 0)u(0) + K(1, 1)u(1)]
Must be solved when the Trapezoidal rule is used
b. Set up and solve the linear system that results when the Composite Trapezoidal rule is used with n = 4.
c. Repeat part (b) using the Composite Simpson s rule. 
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m52516  A function f: [a, b] → R is said to satisfy a Lipschitz condition with Lipschitz constant L on [a, b] if, for every x, y ∈ [a, b], we have f (x) − f (y) ≤ Lx − y.
a. Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an interval [a, b], then f ∈ C[a, b].
b. Showthat if f has a derivative that is bounded on [a, b] by L, then f satisfies a Lipschitz condition with Lipschitz constant L on [a, b].
c. Give an example of a function that is continuous on a closed interval but does not satisfy a
Lipschitz condition on the interval 
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m52524  a. Generalize Cramer s rule to an n × n linear system.
b. Use the result in Exercise 9 to determine the number of multiplications/divisions and additions/subtractions required for Cramer s rule on an n × n system. 
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m52548  a. How many multiplications and additions are required to determine a sum of the form
b. Modify the sum in part (a) to an equivalent form that reduces the number of computations. 
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m52585  A linear dynamical system can be represented by the equations
Dx / dt = A(t)x(t) + B(t)u(t), y(t) = C(t)x(t) + D(t)u(t),
where A is an n × n variable matrix, B is an n × r variable matrix, C is an m × n variable matrix, D is an m × r variable matrix, x is an ndimensional vector variable, y is an mdimensional vector variable, and u is an rdimensional vector variable. For the system to be stable, the matrix A must have all its eigenvalues with nonpositive real part for all t. Is the system stable if
a.
b. 
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m52636  a. Perform four iterations, if possible, on each of the functions g defined in Exercise 1. Let p0 = 1 and pn+1 = g(pn), for n = 0, 1, 2, 3. 
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m52640  A persymmetric matrix is a matrix that is symmetric about both diagonals; that is, an N × N matrix A = (aij) is persymmetric if aij = aji = aN+1−i,N+1−j, for all i = 1, 2, . . . , N and j = 1, 2, . . . , N. A number of problems in communication theory have solutions that involve the eigenvalues and eigenvectors of matrices that are in persymmetric form. For example, the eigenvector corresponding to the minimal eigenvalue of the 4 × 4 persymmetric matrix
gives the unit energychannel impulse response for a given error sequence of length 2, and subsequently the minimum weight of any possible error sequence.
a. Use the Geršgorin Circle Theorem to show that if A is the matrix given above and λ is its minimal eigenvalue, then λ − 4 = ρ(A − 4I), where ρ denotes the spectral radius.
b. Find the minimal eigenvalue of the matrix A by finding all the eigenvalues A−4I and computing its spectral radius. Then find the corresponding eigenvector.
c. Use the Geršgorin Circle Theorem to show that if λ is the minimal eigenvalue of the matrix
then λ − 6 = ρ(B − 6I).
d. Repeat part (b) using the matrix B and the result in part (c). 
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