Given the following function: f (x, y) = 8x + x2 + 12y + 4y2 2xy Determine the minimum

Perform three iterations of the Newton-Raphson method to determine the root of Eq. (E7.1.1). Use the parameter values from Example 7.1 along with an initial guess of t = 3 s.

Given the formula f (x) = x2 + 8x 12 (a) Determine the maximum and the corresponding value of x for this function analytically (i.e., using differentiation). (b) Verify that Eq. (7.10) yields the same results based on initial guesses of x1 = 0, x2 = 2, and x3 = 6.

Consider the following function: f (x) = 3 + 6x + 5x2 + 3x3 + 4x4 Locate the minimum by finding the root of the derivative of this function. Use bisection with initial guesses of xl = 2 and xu = 1.

Given f (x) = 1.5x6 2x4 + 12x (a) Plot the function. (b) Use analytical methods to prove that the function is concave for all values of x. (c) Differentiate the function and then use a root-location method to solve for the maximum f(x) and the corresponding value of x

Solve for the value of x that maximizes f(x) in Prob. 7.4 using the golden-section search. Employ initial guesses of xl = 0 and xu = 2, and perform three iterations.

Repeat Prob. 7.5, except use parabolic interpolation. Employ initial guesses of x1 = 0, x2 = 1, and x3 = 2, and perform three iterations.

Employ the following methods to find the maximum of f (x) = 4x 1.8x2 + 1.2x3 0.3x4 (a) Golden-section search (xl = 2, xu = 4, s = 1%). (b) Parabolic interpolation (x1 = 1.75, x2 = 2, x3 = 2.5, iterations = 5).

Consider the following function: f (x) = x4 + 2x3 + 8x2 + 5x Use analytical and graphical methods to show the function has a minimum for some value of x in the range 2 x

Employ the following methods to find the minimum of the function from Prob. 7.8: (a) Golden-section search (xl = 2, xu = 1, s = 1%). (b) Parabolic interpolation (x1 = 2, x2 = 1, x3 = 1, iterations = 5).

Develop a single script to (a) generate contour and mesh subplots of the following temperature field in a similar fashion to Example 7.4: T (x, y) = 2x2 + 3y2 4xy y 3x and (b) determine the minimum with fminsearch.

The head of a groundwater aquifer is described in Cartesian coordinates by h(x, y) = 1 1 + x2 + y2 + x + xy Develop a single script to (a) generate contour and mesh subplots of the function in a similar fashion to Example 7.4, and (b) determine the maximum with fminsearch.

Recent interest in competitive and recreational cycling has meant that engineers have directed their skills toward the design and testing of mountain bikes (Fig. P7.13a). Suppose that you are given the task of predicting the horizontal and vertical displacement of a bike bracketing system in response to a force. Assume the forces you must analyze can be simplified as depicted in Fig. P7.13b. You are interested in testing the response of the truss to a force exerted in any number of directions designated by the angle . The parameters for the problem are E = Youngs modulus = 2 1011 Pa, A = cross-sectional area = 0.0001 m2 , w = width = 0.44 m,  = length = 0.56 m, and h = height = 0.5 m. The displacements x and y can be solved by determining the values that yield a minimum potential energy. Determine the displacements for a force of 10,000 N and a range of s from 0 (horizontal) to 90 (vertical).

As electric current moves through a wire (Fig. P7.14), heat generated by resistance is conducted through a layer of insulation and then convected to the surrounding air. The steady-state temperature of the wire can be computed as T = Tair + q 2 1 k lnrw + ri rw  + 1 h 1 rw + ri Determine the thickness of insulation ri(m) that minimizes the wires temperature given the following parameters: q = heat generation rate = 75 W/m, rw = wire radius = 6 mm, k = thermal conductivity of insulation = 0.17 W/(m K), h = convective heat transfer coefficient = 12 W/(m2 K), and Tair = air temperature = 293 K.

Develop an M-file that is expressly designed to locate a maximum with the golden-section search. In other words, set if up so that it directly finds the maximum rather than finding the minimum of f(x). The function should have the following features: Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. Return both the optimal x and f(x). Test your program with the same problem as Example 7.1.

Develop an M-file to locate a minimum with the golden-section search. Rather than using the maximum iterations and Eq. (7.9) as the stopping criteria, determine the number of iterations needed to attain a desired tolerance. Test your function by solving Example 7.2 using Ea,d = 0.0001.

Develop an M-file to implement parabolic interpolation to locate a minimum. The function should have the following features: Base it on two initial guesses, and have the program generate the third initial value at the midpoint of the interval. Check whether the guesses bracket a maximum. If not, the function should not implement the algorithm, but should return an error message. Iterate until the relative error falls below a stopping criterion or exceeds a maximum number of iterations. Return both the optimal x and f(x). Test your program with the same problem as Example 7.3.

Pressure measurements are taken at certain points behind an airfoil over time. These data best fit the curve y = 6 cos x 1.5 sin x from x = 0 to 6 s. Use four iterations of the golden-search method to find the minimum pressure. Set xl = 2 and xu = 4.

The trajectory of a ball can be computed with y = (tan 0)x g 2v2 0 cos2 0 x2 + y0 where y = the height (m), 0 = the initial angle (radians), v0 = the initial velocity (m/s), g = the gravitational constant = 9.81 m/s2 , and y0 = the initial height (m). Use the golden-section search to determine the maximum height given y0 = 1 m, v0 = 25 m/s, and 0 = 50. Iterate until the approximate error falls below s = 1% using initial guesses of xl = 0 and xu = 60 m.

A object with a mass of 90 kg is projected upward from the surface of the earth at a velocity of 60 m/s. If the object is subject to linear drag (c = 15 kg/s), use the goldensection search to determine the maximum height the object attains.

The normal distribution is a bell-shaped curve defined by y = ex2 Use the golden-section search to determine the location of the inflection point of this curve for positive x.

Use the fminsearch function to determine the minimum of f (x, y) = 2y2 2.25xy 1.75y + 1.5x2

Use the fminsearch function to determine the maximum of f (x, y) = 4x + 2y + x2 2x4 + 2xy 3y2

Given the following function: f (x, y) = 8x + x2 + 12y + 4y2 2xy Determine the minimum (a) graphically, (b) numerically with the fminsearch function, and (c) substitute the result of (b) back into the function to determine the minimum f (x, y).

The specific growth rate of a yeast that produces an antibiotic is a function of the food concentration c: g = 2c 4 + 0.8c + c2 + 0.2c3 As depicted in Fig. P7.26, growth goes to zero at very low concentrations due to food limitation. It also goes to zero at high concentrations due to toxicity effects. Find the value of c at which growth is a maximum.

A compound A will be converted into B in a stirred tank reactor. The product B and unreacted A are purified in a separation unit. Unreacted A is recycled to the reactor. A process engineer has found that the initial cost of the system is a function of the conversion xA. Find the conversion that will result in the lowest cost system. C is a proportionality constant. Cost = C  1 (1 xA)2 0.6 + 6  1 xA 0.6

A finite-element model of a cantilever beam subject to loading and moments (Fig. P7.28) is given by optimizing f (x, y) = 5x2 5xy + 2.5y2 x 1.5y where x = end displacement and y = end moment. Find the values of x and y that minimize f(x, y).

The Streeter-Phelps model can be used to compute the dissolved oxygen concentration in a river below a point discharge of sewage (Fig. P7.29), o = os kd Lo kd + ks ka  eka t e(kd+ks)t  Sb ka (1 eka t ) (P7.29) where o = dissolved oxygen concentration (mg/L), os = oxygen saturation concentration (mg/L), t = travel time (d), Lo = biochemical oxygen demand (BOD) concentration at the mixing point (mg/L), kd = rate of decomposition of BOD (d1 ), ks = rate of settling of BOD (d1 ), ka = reaeration rate (d1 ), and Sb = sediment oxygen demand (mg/L/d). As indicated in Fig. P7.29, Eq. (P7.29) produces an oxygen sag that reaches a critical minimum level oc, some travel time tc below the point discharge. This point is called critical because it represents the location where biota that depend on oxygen (like fish) would be the most stressed. Determine the critical travel time and concentration, given the following values: os = 10 mg/L kd = 0.1 d1 ka = 0.6 d1 ks = 0.05 d1 Lo = 50 mg/L Sb = 1 mg/L/d

A total charge Q is uniformly distributed around a ringshaped conductor with radius a. A charge q is located at a distance x from the center of the ring (Fig. P7.31). The force exerted on the charge by the ring is given by F = 1 4e0 qQx (x2 + a2)3/2 where e0 = 8.85 1012 C2 /(N m2 ), q = Q = 2 105 C, and a = 0.9 m. Determine the distance x where the force is a maximum.

The torque transmitted to an induction motor is a function of the slip between the rotation of the stator field and the rotor speed s, where slip is defined as s = n nR n where n = revolutions per second of rotating stator speed and nR = rotor speed. Kirchhoffs laws can be used to show that the torque (expressed in dimensionless form) and slip are related by T = 15s(1 s) (1 s)(4s2 3s + 4) Figure P7.32 shows this function. Use a numerical method to determine the slip at which the maximum torque occurs.

The total drag on an airfoil can be estimated by D = 0.01 V2 + 0.95 W V 2 Friction Lift where D = drag, = ratio of air density between the flight altitude and sea level, W = weight, and V = velocity. As seen in Fig. P7.33, the two factors contributing to drag are affected differently as velocity increases. Whereas friction drag increases with velocity, the drag due to lift decreases. The combination of the two factors leads to a minimum drag. ( (a) If = 0.6 and W = 16,000, determine the minimum drag and the velocity at which it occurs. (b) In addition, develop a sensitivity analysis to determine how this optimum varies in response to a range of W = 12,000 to 20,000 with = 0.6.

Roller bearings are subject to fatigue failure caused by large contact loads F (Fig. P7.34). The problem of finding the location of the maximum stress along the x axis can be shown to be equivalent to maximizing the function: f (x) = 0.4 1 + x2  1 + x2 1 0.4 1 + x2 + x Find the x that maximizes f(x). 7

In a similar fashion to the case study described in Sec. 7.4, develop the potential energy function for the system depicted in Fig. P7.35. Develop contour and surface plots in MATLAB. Minimize the potential energy function to determine the equilibrium displacements x1 and x2 given the forcing function F = 100 N and the parameters ka = 20 and kb = 15 N/m.

As an agricultural engineer, you must design a trapezoidal open channel to carry irrigation water (Fig. P7.36). Determine the optimal dimensions to minimize the wetted perimeter for a cross-sectional area of 50 m2 . Are the relative dimensions universal?

Use the function fminsearch to determine the length of the shortest ladder that reaches from the ground over the fence to the buildings wall (Fig. P7.37). Test it for the case where h = d = 4 m.

The length of the longest ladder that can negotiate the corner depicted in Fig. P7.38 can be determined by computing the value of that minimizes the following function: L() = w1 sin + w2 sin( ) For the case where w1 = w2 = 2 m, use a numerical method described in this chapter (including MATLABs built-in capabilities) to develop a plot of L versus a range of s from 45 to 135.