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mathematics
numerical analysis

Questions and Answers of Numerical Analysis

As the previous example demonstrated, utility maximization places no restrictions on the slopes of individual demand functions. However, the consumer's demand must always satisfy the budget
In preceding example show that the net revenue functionis supermodular in x and displays strictly increasing differences in (x, - w) provided that the production function f is supermodular. In the
Show that the objective function in the preceding example displays increasing differences in (p, θ).
Apply the conservation of volume (see Prob. 1.9) to simulate the level of liquid in a conical storage tank (Fig. P1.11). The liquid flows in at a sinusoidal rate of Qin = 3 sin2(t) and flows out
As depicted in Fig. P1.15, an RLC circuit consists of three elements: a resistor (R), and inductor (L) and a capacitor (C). The flow of current across each element induces a voltage drop.
The velocity is equal to the rate of change of distance x (m),(a) Substitute Eq. (1.10) and develop an analytical solution for distance as a function of time. Assume that x(0) = 0. (b) Use Euler's
You are working as a crime-scene investigator and must predict the temperature of a homicide victim over a 5-hr period. You know that the room where the victim was found was at 10∘C when the body
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)
As noted in Prob. 1.3, drag is more accurately represented as depending on the square of velocity. A more fundamental representation of the drag force, which assumes turbulent conditions (i.e., a
As depicted in Fig. P1.22, a spherical particle settling through a quiescent fluid is subject to three forces: the downward force of gravity (FG), and the upward forces of buoyancy (FB) and drag
As described in Prob. 1.22, in addition to the downward force of gravity (weight) and drag, an object falling through a fluid is also subject to a buoyancy force that is proportional to the displaced
As depicted in Fig. P1.24, the downward deflection y (m) of a cantilever beam with a uniform load w (kg/m) can be computed asWhere x = distance (m), E = the modulus of elasticity = 2 ×1011 Pa, I=
Use Archimedes' principle to develop a steady-state force balance for a spherical ball of ice floating in seawater (Fig. P1.25). The force balance should be expressed as a third-order polynomial
Beyond fluids, Archimedes' principle has proven useful in geology when applied to solids on the earth's crust. Figure P1.26 depicts one such case where a lighter conical granite mountain "floats on"
The following information is available for a bank account:Note that the money earns interest which is computed as Interest = iBi where i = the interest rate expressed as a fraction per month, and Bi
A group of 35 students attend a class in a room that measures 11 m by 8 m by 3 m. Each student takes up about 0.075 m3 and gives out about 80 W of heat (1 W = 1 J/s). Calculate the air temperature
Develop well-structured function procedures to determine (a) The factorial; (b) The minimum value in a vector; and (c) The average of the values in a vector.
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
Develop a well-structured function to determine the elapsed days in a year. The function should be passed three values: mo = the month (1-12), da = the day (1-31) and leap = (0 for non-leap year and
Develop a well-structured function to determine the elapsed days in a year. The first line of the function should be set up as function nd = days(mo, da, year) where mo = the month (1-12), da = the
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 pseudocode in Fig. P2.25 computes the factorial. Express this algorithm as a well-structured function in the language of your choice. Test it by computing 0! and 5!. In addition, test the error
The height of a small rocket y can be calculated as a function of time after blastoff with the following piecewise function: Y = 38.1454t + 0.13743t3 0 ≤ t < 15 y = 1036 + 130.909(t - 15)
As depicted in Fig. P2.27, a water tank consists of a cylinder topped by the frustum of a cone. Develop a well structured function in the high-level language or macro language of your choice to
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
Use bisection to determine the drag coefficient needed so that an 82-kg parachutist has a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s2. Start with initial
As depicted in Fig. P5.15, the velocity of water, y (m/s), discharged from a cylindrical tank through a long pipe can be computed aswhere g = 9.81 m/s2, H = initial head (m), L = pipe length (m),
According to Archimedes principle, the buoyancy force is equal to the weight of fl uid displaced by the submerged portion of an object. For the sphere depicted in Fig. P5.19, use bisection to
You must determine the root of the following easily differentiable function, e0.5x = 5 - 5x Pick the best numerical technique, justify your choice and then use that technique to determine the root.
(a) Apply the Newton-Raphson method to the function f(x) = tanh(x2 - 9) to evaluate its known real root at x = 3. Use an initial guess of x0 = 3.2 and take a minimum of four iterations. (b) Did the
Develop a user-friendly program for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Test it by duplicating the computation from Example 6.3.
Develop a user-friendly program for Brent's root location method based on Fig. 6.12. Test it by solving Prob. 6.6.
Develop a user-friendly program for the two-equation Newton-Raphson method based on Sec. 6.6.2. Test it by solving Example 6.12.
Use MATLAB or Mathcad to determine the roots for the equations in Prob. 7.5.
Develop an M-file function for the Newton-Raphson method based on Fig. 6.4 and Sec. 6.2.3. Along with the initial guess, pass the function and its derivative as arguments. Test it by duplicating the
The Manning equation can be written for a rectangular open channel aswhere Q = flow [m3/s], S 5 slope [m/m], H = depth [m], and n = the Manning roughness coefficient. Develop a fixed-point iteration
You buy a $20,000 piece of equipment for nothing down and $4000 per year for 6 years. What interest rate are you paying? The formula relating present worth P, annual payments A, number of years n,
Although we did not mention it in Sec. 8.2, Eq. (8.10) is actually an expression of electro neutrality; that is, that positive and negative charges must balance. This can be seen more clearly by
Beyond the Colebrook equation, other relationships, such as the Fanning friction factor f, are available to estimate friction in pipes. The Fanning friction factor is dependent on a number of
The general form for a three-dimensional stress field is given bywhere the diagonal terms represent tensile or compressive stresses and the off-diagonal terms represent shear stresses. A stress field
Determining the velocity of particles settling through fluids is of great importance of many areas of engineering and science. Such calculations depend on the flow regime as represented by the
The Redlich-Kwong equation of state is given bywhere R = the universal gas constant [= 0.518 kJ/(kg K)], T = absolute temperature (K), p = absolute pressure (kPa), and v = the volume of a kg of gas
Use Gauss-Jordan elimination to solve: 2x1 + x2 - x3 = 1 5x1 + 2x2 + 2x3 = 24 3x1 + x2 + x3 = 5 Do not employ pivoting. Check your answers by substituting them into the original equations.
Three masses are suspended vertically by a series of identical springs where mass 1 is at the top and mass 3 is at the bottom. If g = 9.81 m/s2, m1 = 2 kg, m2 = 3 kg, m3 = 2.5 kg, and the k's = 10
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],
Use Gauss elimination to solve:8x1 + 2x2 – 2x3 = 2210x1 + 2x2 + 4x3 = 412x1 + 2x2 + 2x3 = 6Employ partial pivoting and check your answers by substituting them into the original equations.
Determine the Frobenius and the row-sum norms for the systems in Probs. 10.3 and 10.4. Scale the matrices by making the maximum element in each row equal to one?
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?
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]?
As described in Sec. PT3.1.2, linear algebraic equations can arise in the solution of differential equations. For example, the following differential equation results from a steady-state mass balance
Use the golden-section search to determine the length of the shortest ladder that reaches from the ground over the fence to touch the building's wall (Fig. P13.22). Test it for the case where h= d =
Develop an M-file that is expressly designed to locate a maximum with the golden-section search algorithm. In other words, set it up so that it directly finds the maximum rather than finding the
Develop an M-file to locate a minimum with the golden section search. Rather than using the standard stopping criteria (as in Fig. 13.5), determine the number of iterations needed to attain a desired
Develop an M-file to implement parabolic interpolation to locate a minimum. Test your program with the same problem as Example 13.2. The function should have the following features: • Base it on
The length of the longest ladder that can negotiate the corner depicted in Fig. P15.17 can be determined by computing the value of u that minimizes the following function:For the case where w1 = w2 =
Design the optimal cylindrical tank with dished ends (Fig. P16.3). The container is to hold 0.5 m3 and has walls of negligible thickness. Note that the area and volume of each of the dished ends can
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
Recent interest in competitive and recreational cycling has meant that engineers have directed their skills toward the designFigure P16.32Two frictionless masses connected to a wall by a pair of
Given these dataDetermine (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
Given these dataDetermine(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
Use Newton's interpolating polynomial to determine y at x = 8 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy
The following is the built-in humps function that MATLAB uses to demonstrate some of its numerical capabilities:The humps function exhibits both fl at and steep regions over a relatively short x
The following data define the sea-level concentration of dissolved oxygen for fresh water as a function of temperature:Estimate o(27) using (a) Linear interpolation, (b) Newton's interpolating
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?
The following data come from a table that was measured with high precision. Use the best numerical method (for this type of problem) to determine y at x = 3.5. Note that a polynomial will yield an
Use Newton's interpolating polynomial to determine y at x = 3.5 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy
The data tabulated below were generated from an experiment initially containing pure ammonium cyanate (NH4OCN). It is known that such concentration changes can be modeled by the following equation:c
The following model is frequently used in environmental engineering to parameterize the effect of temperature T (°C) on biochemical reaction rates k (per day),k = k20θT–20where k20 and u are
As a member of Engineers Without Borders, you are working in a community that has contaminated drinking water. At t = 0, you add a disinfectant to a cistern that is contaminated with bacteria. You
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
The following data was collected for a cross-section of a river (y = distance from bank, H = depth, and U = velocity):Use numerical integration to compute the (a) average depth, (b) cross-sectional
Develop a user-friendly computer program for adaptive quadrature based on Fig. 22.5. Test it by solving Prob. 22.10.Fig 22.5.
Prove that Eq. (22.15) is equivalent to Boole's rule.Eq . 22.15
Employ two- through six-point Gauss-Legendre formulas to solveInterpret your results in light of Eq. (22.32)
The following data were collected for the distance traveled versus time for a rocket:Use numerical differentiation to estimate the rocket's velocity and acceleration at each time.
The following relationships can be used to analyze uniform beams subject to distributed loads,Where x = distance along beam (m), y = deflection (m), u(x) = slope (m/m), E = modulus of elasticity (Pa
You measure the following deflections along the length of a simply-supported uniform beamEmploy numerical differentiation to compute the slope, the moment (in N m), the shear (in N) and the
Compute work as described in Sec. 24.4, but use the following equations for F(x) and θ(x): F(x) = 1.6x - 0.045x2 u(x) = -0.00055x3 + 0.0123x2 + 0.13x The force is in newtons and the angle is in
As was done in Sec. 24.4, determine the work performed if a constant force of 1 N applied at an angle u results in the following displacements. Use the MATLAB function cumtrapz to determine the
Integration provides a means to compute how much mass enters or leaves a reactor over a specified time period, as inWhere t1 and t2 = the initial and final times, respectively. This formula makes
Use numerical integration to compute how much mass leaves a reactor based on the following measurements.
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
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
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
Use the following differential equations to compute the velocity and position of a soccer ball that is kicked straight up in the air with an initial velocity of 40 m/s:Where y = upward distance (m),
Three linked bungee jumpers are depicted in Fig. P25.26. If the bungee cords are idealized as linear springs (i.e., governed by Hooke's law), the following differential equations based on force
Use (a) Euler's and (b) the fourth-order RK method to solveOver the range t = 0 to 0.4 using a step size of 0.1 with y(0) = 2 and z(0) = 4.
Use MATLAB or Mathcad to integrate dx/dt = -θx + θy dy/dt = rx - y - xz dz/dt = -bz + xy Where s = 10, b = 2.666667, and r = 28. Employ initial conditions of x = y = z = 5 and integrate from t = 0
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.
A biofilm with a thickness Lf (cm) grows on the surface of a solid. After traversing a diffusion layer of thicknessL (cm), a chemical compound A diffuses into the biofilm where it is subject to an
The following differential equation describes the steady state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactorWhere D = the dispersion
A series of first-order, liquid-phase reactions create a desirable product (B) and an undesirable byproduct (C)If the reactions take place in an axially-dispersed plug-flow reactor (Fig. P28.14),
Bacteria growing in a batch reactor utilize a soluble food source (substrate) as depicted in Fig. P28.16. The uptake of the substrate is represented by a logistic model with Michaelis-Menten
Under a number of simplifying assumptions, the steadystate height of the water table in a one-dimensional, unconfined groundwater aquifer (Fig. P28.30) can be modeled with the following second-order
A linearized groundwater model was used to simulate the height of the water table for an unconfined aquifer. A more realistic result can be obtained by using the following nonlinear ODE:Where x =
The Lotka-Volterra equations described in Sec. 28.2 have been refined to include additional factors that impact predator-prey dynamics. For example, over and above predation, prey population can be
The growth of floating, unicellular algae below a sewage treatment plant discharge can be modeled with the following simultaneous ODEs:Where t = travel time (d), a = algal chlorophyll concentration
The following ODEs have been proposed as a model of an epidemic:Where S = the susceptible individuals, I = the infected, R = the recovered, a = the infection rate, and r = the recovery rate. A city
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
Two masses are attached to a wall by linear springs. Force balances based on Newton's second law can be written asWhere k = the spring constants, m = mass, L = the length of the unstretched spring,

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