Numerical Analysis 9th Edition solutions

Numerical Analysis 9
Use the Composite Trapezoidal rule with the indicated values of n to approximate the following integrals. a. 2 1 x ln x dx, n = 4 b. 2 2 x3 ex dx, n = 4 c. 2 0 2 x2 + 4 dx, n = 6 d. 0 x2 cos x dx, n = 6 e. 2 0 e2x sin 3x dx, n = 8 f. 3 1 x x2 + 4 dx, n = 8 g. 5 3 1 x2 4 dx, n = 8 h. 3/8 0 tan x dx,

Numerical Analysis 9
Use the Composite Trapezoidal rule with the indicated values of n to approximate the following integrals. a. 0.5 0.5 cos2 x dx, n = 4 b. 0.5 0.5 x ln(x + 1) dx, n = 6 c. 1.75 .75 (sin2 x 2x sin x + 1) dx, n = 8 d. e+2 e 1 x ln x dx, n = 8 3.

Numerical Analysis 9
Use the Composite Simpsons rule to approximate the integrals in Exercise 1.

Numerical Analysis 9
Use the Composite Simpsons rule to approximate the integrals in Exercise 2.

Numerical Analysis 9
Use the Composite Midpoint rule with n + 2 subintervals to approximate the integrals in Exercise 1.

Numerical Analysis 9
Use the Composite Midpoint rule with n + 2 subintervals to approximate the integrals in Exercise 2.

Numerical Analysis 9
Approximate  2 0 x2 ln(x2 + 1) dx using h = 0.25. Use a. Composite Trapezoidal rule. b. Composite Simpsons rule. c. Composite Midpoint rule.

Numerical Analysis 9
Approximate  2 0 x2ex2 dx using h = 0.25. Use a. Composite Trapezoidal rule. b. Composite Simpsons rule. c. Composite Midpoint rule.

Numerical Analysis 9
Suppose that f (0) = 1, f (0.5) = 2.5, f (1) = 2, and f (0.25) = f (0.75) = . Find if the Composite Trapezoidal rule with n = 4 gives the value 1.75 for  1 0 f (x) dx.

Numerical Analysis 9
The Midpoint rule for approximating  1 1 f (x) dx gives the value 12, the Composite Midpoint rule with n = 2 gives 5, and Composite Simpsons rule gives 6. Use the fact that f (1) = f (1) and f (0.5) = f (0.5) 1 to determine f (1), f (0.5), f (0), f (0.5), and f (1).

Numerical Analysis 9
Determine the values of n and h required to approximate 2 0 e2x sin 3x dx to within 104. Use a. Composite Trapezoidal rule. b. Composite Simpsons rule. c. Composite Midpoint rule.

Numerical Analysis 9
Repeat Exercise 11 for the integral  0 x2 cos x dx.

Numerical Analysis 9
Determine the values of n and h required to approximate 2 0 1 x + 4 dx to within 105 and compute the approximation. Use a. Composite Trapezoidal rule. b. Composite Simpsons rule. c. Composite Midpoint rule.

Numerical Analysis 9
Repeat Exercise 13 for the integral  2 1 x ln x dx.

Numerical Analysis 9
Let f be defined by f (x) = x3 + 1, 0 x 0.1, 1.001 + 0.03(x 0.1) + 0.3(x 0.1)2 + 2(x 0.1)3, 0.1 x 0.2, 1.009 + 0.15(x 0.2) + 0.9(x 0.2)2 + 2(x 0.2)3, 0.2 x 0.3. a. Investigate the continuity of the derivatives of f . b. Use the Composite Trapezoidal rule with n = 6 to approximate  0.3

Numerical Analysis 9
Show that the error E(f ) for Composite Simpsons rule can be approximated by h4 180 [f (b) f (a)]. [Hint: n/2 j=1 f (4) (j)(2h) is a Riemann Sum for  b a f (4) (x) dx.]

Numerical Analysis 9
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.

Numerical Analysis 9
Use the error estimates of Exercises 16 and 17 to estimate the errors in Exercise 12.

Numerical Analysis 9
Use the error estimates of Exercises 16 and 17 to estimate the errors in Exercise 14.

Numerical Analysis 9
In multivariable calculus and in statistics courses it is shown that 1 2 e(1/2)(x/ )2 dx = 1, for any positive . The function f (x) = 1 2 e(1/2)(x/ )2 is the normal density function with mean = 0 and standard deviation . The probability that a randomly chosen value described by this distribution

Numerical Analysis 9
Determine to within 106 the length of the graph of the ellipse with equation 4x2 + 9y2 = 36.

Numerical Analysis 9
A car laps a race track in 84 seconds. The speed of the car at each 6-second interval is determined by using a radar gun and is given from the beginning of the lap, in feet/second, by the entries in the following table. Time 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 Speed 124 134 148 156 147 133 121

Numerical Analysis 9
A particle of mass m moving through a fluid is subjected to a viscous resistance R, which is a function of the velocity v. The relationship between the resistance R, velocity v, and time t is given by the equation t = v(t) v(t0) m R(u) du. Suppose that R(v) = v v for a particular fluid, where R is in

Numerical Analysis 9
To simulate the thermal characteristics of disk brakes (see the following figure), D. A. Secrist and R. W. Hornbeck [SH] needed to approximate numerically the area averaged lining temperature, T, of the brake pad from the equation T = r0 re T(r)rp dr r0 re rp dr , where re represents the radius at wh

Numerical Analysis 9
Find an approximation to within 104 of the value of the integral considered in the application opening this chapter: 48 0  1 + (cos x)2 dx.

Numerical Analysis 9
The equation x 0 1 2 et 2/2 dt = 0.45 can be solved for x by using Newtons method with f (x) = x 0 1 2 et 2/2 dt 0.45 and f (x) = 1 2 ex2/2 . To evaluate f at the approximation pk , we need a quadrature formula to approximate pk 0 1 2 et 2/2 dt. a. Find a solution to f (x) = 0 accurate to within 1