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mathematics
numerical analysis

Questions and Answers of Numerical Analysis

A transportation engineering study requires the calculation of the total number of cars that pass through an intersection over a 24-h period. An individual visits the intersection at various times
A wind force distributed against the side of a skyscraper is measured asCompute the net force and the line of action due to this distributedwind.
Water exerts pressure on the upstream face of a dam as shown in Figure. The pressure can be characterized byp(z) = pg(D - z)Where p(z) = pressure in pascals (or N/m2) exerted at an elevation z meters
To estimate the size of a new dam, you have to determine the total volume of water (m3) that flows down a river in a year’s time. You have available the following long-term average data for the
The data listed in the following table gives hourly measurements of heat flux q (cal/cm2/h) at the surface of a solar collector. As an architectural engineer, you must estimate the total heat
The heat flux q is the quantity of heat flowing through a unit area of a material per unit time. It can be computed with Fourier’s law,Where J has units of J/m2/s or W/m2 and k is a coefficient of
The horizontal surface area As (m2) of a lake at a particular depth can be computed from volume by differentiation,Where V= volume (m3) and z = depth (m) as measured from the surface down to the
Perform the same computation as in Sec. 24.3, but for the current as specified byi(t) = 5e-1.25t sin 2πt for 0 ≤ t ≤ T/2i(t) = 0 for/ T/2 ≤ t ≤ TWhere T = I s. Use five-point Gauss
Repeat Prob. 24.27, but use is five-segment Simpson’s 1/3 rule.
Repeat Prob. 24.27, but use Romberg integration to εs 1%.
Faraday’s law characterizes the voltage drop across an inductor asWhere VL = voltage drop (V), L inductance (in henrys; 1 H = l V ∙ s/A), i = current (A), and t = time (s). Determine the
Based on Fareday’s law (Prob. 24.30), use the following voltage data to estimate the inductance in henrys if a current of 2 A is passed through the inductor over 400 milliseconds.
Suppose that the current through a resistor is described by the functioni(t) = (60 - t)2 + (60 - t) sin(√t)and the resistance is a function of the current,R = 12i + 2i2/3Compute the average voltage
If a capacitor initially holds no charge, the voltage across it as a function of time can be computed asIf C = l0-5 farad, use the following current data to develop a plot of voltage versustime:
Perform the same computation as in Sec. 24.4, but use the following equation:F(x) = 1.6x – 0.045x2Employ the values of θ from Table 24.6.
Perform the same computation as in Sec. 24.4, but use the following equation:θ(x) = 0.8 + 0.125x – 0.009x2 + 0.0002x3Employ the equation from Prob. 24.34 for F(x). Use 4-, 8, and 16-segment
Repeat Prob. 24.35, but use Simpson’s 1/3 rule.
Repeat Prob.24.35, but use Romberg integration to εs= 0.5%.
Repeat Prob. 24.35, bat use Gauss quadrature.
The work done on an object is equal to the force times the distance moved in the direction of the force. The velocity of an object in the direction of a force is given byυ = 4t 0 ≤ t ≤ 4υ =
The rate of cooling of a body (Figure) can be expressed asWhere T = temperature of the body (°C). Tα = temperature of the surrounding medium (°C), and k = a proportionality constant (per minute)
A rod subject to an axial load (Figure) will be deformed, as shown in the stress-Strain curve in Figure. The area under the curve from zero stress out to the point of rupture is called the modulus of
If the velocity distribution of a fluid flowing through a pipe is known (Figure), the flow rate Q (that is, the volume of water passing through the pipe per unit time) can be computed by Q = ∫υdA,
Using the following data, calculate the work done by stretching a spring that has a spring constant of k = 300 N/m to x = 0.35m:
A jet fighter’s position on an aircraft carrier’s runway was timed during landing:where x is the distance from the and of the carrier. Estimate(a) Velocity (dx/dt) and(b) Acceleration (dÏ…/dt)
Employ the multiple-application Simpson’s rule to evaluate the vertical distance traveled by a rocket if the vertical velocity is given byυ = 11t2 – 5t
The upward velocity of a rocket can he computed by the following formula:where υ = upward velocity, u = velocity at which fuel is expelled relative to the rocket, m0 = initial mass of the
Referring to the data from Problem 20.57, find the strain rate using finite difference methods. Use first-order accurate derivative approximations and plot your results. Looking at the graph, it is
Fully developed flow moving through a 40-cm diameter pipe has the following velocity profile:Find the volume flow rate Q using the relationship Q = ∫R0 2πrυ dr, where r is the radial axis of the
Fully developed flow of a Bingham plastic fluid moving through a 12-in diameter pipe has the given velocity profile. The flow of a Bingham fluid does not shear the center core, producing plug flow in
The enthalpy of a real gas is a function of pressure as described below. The data was taken for a real fluid. Estimate the enthalpy of the fluid at 400 K and 50 atm (evaluate the integral from 0 atm
Given the data below, find the isothermal work done on the gas as it is compressed from 23 L to 3 L (remember that W = - ∫v2v1 p dV|).(a) Find the work performed on the gas numerically, using the
The Rosin-Rammler-Bennet (RRB) equation is used to described size distribution in fine dust F(x) represents the cumulative mass of dust particles of diameter x and smaller. x’ and n’ are
For fluid flow over a surface, the heat flux to the surface can be computed asJ = -k dT/dyWhere J = heat flux (W/m2), k = thermal conductivity (W/m ∙ K) T = temperature (K), and y = distance normal
The pressure gradient for laminar flow through a constant radius tube is given bywhere p = pressure (N/m2), x = distance along the tube’s centerline (m), μ = dynamic viscosity (N ∙ s/m2), Q =
Velocity data for air are collected at different radii from the centerline of a circular 16-cm diameter pipe as tabulated below:Use numerical integration to determine the mass flow rate, which can be
The Macon Psychiatric Institute is interested in redesigning its mental health care delivery system in order to maximize the number of people who can benefit from its services. Unfortunately, having
Lawn King manufactures two types of riding lawn mowers. One is a low-cost mower sold primarily to residential home owners; the other is an industrial model sold to landscaping and lawn service
In the Markowitz portfolio optimization model defined in equations (8.10) through (8.19), the decision variables represent the percentage of the portfolio invested in each of the mutual funds. For
The weekly box office revenues (in $ millions) for Terminator 3 are given here. Use these data in the Bass forecasting model given by equations through to estimate the parameters p, q, and m. Solve
Show that the following equations have at least one solution in the given intervals. a. x cos x − 2x2 + 3x − 1 = 0, [0.2, 0.3] and [1.2, 1.3] b. (x − 2)2 − ln x = 0, [1, 2] and [e, 4] c. 2x
Repeat Exercise 9 using x0 = Ï€/6.a. Use P2 (0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) ˆ’ P2(0.5)| using the error formula, and compare it to the actual error.b. Find
Find the third Taylor polynomial P3(x) for the function f (x) = (x ˆ’ 1) ln x about x0 = 1.a. Use P3(0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) ˆ’ P3(0.5)| using the
Let f (x) = 2x cos(2x) − (x − 2)2 and x0 = 0. a. Find the third Taylor polynomial P3(x), and use it to approximate f (0.4). b. Use the error formula in Taylor's Theorem to find an upper bound for
Find the fourth Taylor polynomial P4(x) for the function f (x) = x(ex)2 about x0 = 0.a. Find an upper bound for |f (x) ˆ’ P4(x)|, for 0 ‰¤ x ‰¤ 0.4.b. Approximatec. Find an upper bound for
Use the error term of a Taylor polynomial to estimate the error involved in using sin x ≈ x to approximate sin 1◦.
Use a Taylor polynomial about π/4 to approximate cos 42◦ to an accuracy of 10−6.
Let f (x) = ex/2 sin(x/3). Use Maple to determine the following.a. The third Maclaurin polynomial P3(x).b. f (4)(x) and a bound for the error |f (x) − P3(x)| on [0, 1].
Let f (x) = ln(x2 + 2). Use Maple to determine the following.a. The Taylor polynomial P3(x) for f expanded about x0 = 1.b. The maximum error |f (x) − P3(x)|, for 0 ≤ x ≤ 1.c. The Maclaurin
Find intervals containing solutions to the following equations. a. x − 3−x = 0 b. 4x2 − ex = 0 c. x3 − 2x2 − 4x + 2 = 0 d. x3 + 4.001x2 + 4.002x + 1.101 = 0
Find the nth Maclaurin polynomial Pn(x) for f (x) = arctan x.
The nth Taylor polynomial for a function f at x0 is sometimes referred to as the polynomial of degree at most n that "best" approximates f near x0. a. Explain why this description is accurate. b.
Prove the Generalized Rolle's Theorem, Theorem 1.10, by verifying the following. a. Use Rolle's Theorem to show that f'(zi) = 0 for n − 1 numbers in [a, b] with a < z1 < z2 < · · · < zn−1 <
In Example 3 it is stated that for all x we have | sin x| ≤ |x|. Use the following to verify this statement. a. Show that for all x ≥ 0we have f (x) = x−sin x is non-decreasing, which implies
The error function defined bygives the probability that any one of a series of trials will lie within x units of the mean, assuming that the trials have a normal distribution with mean 0 and standard
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)| ≤ L|x − y|. a. Show that if f
Suppose f ˆˆ C[a, b], that x1 and x2 are in [a, b].a. Show that a number Î¾ exists between x1 and x2 withb. Suppose that c1 and c2 are positive constants. Show that a number Î¾ exists between
Let f ∈ C[a, b], and let p be in the open interval (a, b).a. Suppose f (p) ≠ 0. Show that a δ > 0 exists with f (x) ≠ 0, for all x in [p − δ, p + δ], with [p − δ, p + δ] a subset of
Show that f'(x) is 0 at least once in the given intervals. a. f (x) = 1 − ex + (e − 1) sin((π/2)x), [0, 1] b. f (x) = (x − 1) tan x + x sin πx, [0, 1] c. f (x) = x sin πx − (x − 2) ln x,
Find max a≤x≤b |f (x)| for the following functions and intervals a. f (x) = (2 − ex + 2x)/3, [0, 1] b. f (x) = (4x − 3)/(x2 − 2x), [0.5, 1] c. f (x) = 2x cos(2x) − (x − 2)2, [2, 4] d. f
Use the Intermediate Value Theorem 1.11 and Rolle's Theorem 1.7 to show that the graph of f (x) = x3 + 2x + k crosses the x-axis exactly once, regardless of the value of the constant k.
Suppose f ∈ C[a, b] and f'(x) exists on (a, b). Show that if f'(x) ≠ 0 for all x in (a, b), then there can exist at most one number p in [a, b] with f (p) = 0.
Let f (x) = x3. a. Find the second Taylor polynomial P2(x) about x0 = 0. b. Find R2 (0.5) and the actual error in using P2 (0.5) to approximate f (0.5). c. Repeat part (a) using x0 = 1. d. Repeat
Find the third Taylor polynomial P3(x) for the function f (x) = √(x + 1) about x0 = 0. Approximate √0.5, √0.75, √1.25, and √1.5 using P3(x), and find the actual errors.
Find the second Taylor polynomial P2(x) for the function f (x) = ex cos x about x0 = 0.a. Use P2 (0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) ˆ’ P2(0.5)| using the error
Compute the absolute error and relative error in approximations of p by p∗. a. p = π, p∗ = 22/7 b. p = π, p∗ = 3.1416 c. p = e, p∗ = 2.718 d. p =√2, p∗ = 1.414 e. p = e10, p∗ =
The number e can be definedWhere n! = n(nˆ’1) · · · 2 · 1 for n ‰ 0 and 0! = 1. Compute the absolute error and relative error in the following approximations of e:
Leta. Find limx†’0 f (x).b. Use four-digit rounding arithmetic to evaluate f (0.1).c. Replace each trigonometric function with its third Maclaurin polynomial, and repeat part (b).d. The actual
Let f (x) = (ex − e−x)/x. a. Find limx→0 (ex − e−x)/x. b. Use three-digit rounding arithmetic to evaluate f (0.1). c. Replace each exponential function with its third Maclaurin polynomial,
Use four-digit rounding arithmetic and the formulas (1.1), (1.2), and (1.3) to find the most accurate approximations to the roots of the following quadratic equations. Compute the absolute errors and
Use four-digit chopping arithmetic and the formulas (1.1), (1.2), and (1.3) to find the most accurate approximations to the roots of the following quadratic equations. Compute the absolute errors and
Use the 64-bit long real format to find the decimal equivalent of the following floating-point machine numbers. a. 0 10000001010 1001001100000000000000000000000000000000000000000000 b. 1 10000001010
Find the next largest and smallest machine numbers in decimal form for the numbers a. 0 10000001010 1001001100000000000000000000000000000000000000000000 b. 1 10000001010
The Taylor polynomial of degree n forUse the Taylor polynomial of degree nine and three-digit chopping arithmetic to find an approximation to eˆ’5 by each of the following methods.c. An
The two-by-two linear system ax + by = e, cx + dy = f , Where a, b, c, d, e, f are given, can be solved for x and y as follows: Set m = c/a , provided a ≠ 0; d1 = d − mb; f1 = f − me; y =
Find the largest interval in which p∗ must lie to approximate p with relative error at most 10−4 for each value of p a. π b. e c.√2 d. 3√7
The two-by-two linear system ax + by = e, cx + dy = f , Where a, b, c, d, e, f are given, can be solved for x and y as follows: Set m = c/a, provided a ≠ 0; d1 = d − mb; f1 = f − me; y =
a. Show that the polynomial nesting technique described in Example 6 can also be applied to the evaluation of f (x) = 1.01e4x − 4.62e3x − 3.11e2x + 12.2ex − 1.99. b. Use three-digit rounding
A rectangular parallelepiped has sides of length 3 cm, 4 cm, and 5 cm, measured to the nearest centimeter. What are the best upper and lower bounds for the volume of this parallelepiped? What are the
Suppose that f l(y) is a k-digit rounding approximation to y. Show that |y − fl(y)/y| ≤ 0.5 × 10−k+1. [If dk+1 < 5, then f l(y) = 0.d1d2 . . . dk ×10n. If dk+1 ≥ 5, then f l(y) = 0.d1d2
The binomial coefficient (m/k) = m!/(k! (m− k)!) Describes the number of ways of choosing a subset of k objects from a set of m elementsa. Suppose decimal machine numbers are of the form
Let f ∈ C[a, b] be a function whose derivative exists on (a, b). Suppose f is to be evaluated at x0 in (a, b), but instead of computing the actual value f (x0), the approximate value, (x0), is
The following Maple procedure chops a floating-point number x to t digits. (Use the Shift and Enter keys at the end of each line when creating the procedure.) chop := proc(x, t); local e, x2; if x =
The opening example to this chapter described a physical experiment involving the temperature of a gas under pressure. In this application, we were given P = 1.00 atm, V = 0.100 m3, N = 0.00420 mol,
Suppose p∗ must approximate p with relative error at most 10−3. Find the largest interval in which p∗ must lie for each value of p a. 150 b. 900 c. 1500 d. 90
Perform the following computations (i) exactly, (ii) using three-digit chopping arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Compute the relative errors in parts (ii) and
Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits.a. 133 + 0.921b. 133
Use four-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits.a. 133 + 0.921b. 133
Use three-digit chopping arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits.a. 133 + 0.921b. 133
Use four-digit chopping arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits.a. 133 + 0.921b. 133
The first three nonzero terms of the Maclaurin series for the arctangent function are x − (1/3)x3+ (1/5)x5. Compute the absolute error and relative error in the following approximations of π using
a. Use three-digit chopping arithmetic to compute the sumfirst by 1/1 + 1/4 +· · ·+ 1/100 and then by 1/100 + 1/81+· · ·+ 1/1 . Which method is more accurate, and why?b. Write an algorithm to
Equations (1.2) and (1.3) in Section 1.2 give alternative formulas for the roots x1 and x2 of ax2 + bx + c = 0. Construct an algorithm with input a, b, c and output x1, x2 that computes the roots x1
Construct an algorithm that has as input an integer n ≥ 1, numbers x0, x1, . . . , xn, and a number x and that produces as output the product (x − x0)(x − x1) · · · (x − xn).
Assume that (1−2x)/(1−x+x2) + (2x−4x3)/(1−x2+x4) + (4x3−8x7)/(1 − x4+x8) +· · · = (1 + 2x)/(1+x+x2), for x < 1, and let x = 0.25. Write and execute an algorithm that determines the
a. Suppose that 0 < q < p and that αn = α + O(n−p). Show that αn = α + O(n−q). b. Make a table listing 1/n, 1/n2, 1/n3, and 1/n4 for n = 5, 10, 100, and 1000, and discuss the varying rates of
a. Suppose that 0 < q < p and that F(h) = L + O(hp). Show that F(h) = L + O(hq). b. Make a table listing h, h2, h3, and h4 for h = 0.5, 0.1, 0.01, and 0.001, and discuss the varying rates of
Suppose that as x approaches zero, F1(x) = L1 + O(xα) and F2(x) = L2 + O(xβ). Let c1 and c2 be nonzero constants, and define F(x) = c1F1(x) + c2F2(x) and G(x) = F1(c1x) + F2(c2x). Show that if γ =
The sequence {Fn} described by F0 = 1, F1 = 1, and Fn+2 = Fn+Fn+1, if n ≥ 0, is called a Fibonacci sequence. Its terms occur naturally in many botanical species, particularly those with petals or
The Fibonacci sequence also satisfies the equation Fn ≡ n = 1/√5[(1 +√5/2)n −(1 −√5/2)n]. a. Write a Maple procedure to calculate F100. b. Use Maple with the default value of Digits

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