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1EFind intervals containing solutions to the following equations. x3x=0 4x2ex=0 x32x24x+3=0 x3+4.001x2+4.002x+1.101=03EFind maxaxbf(x) for the following functions and intervals. f(x)=2ex+2x/3,[0,1] f(x)=(4x3)/x22x,[0.5,1] f(x)=2xcos(2x)(x2)2,[2,4] f(x)=1+ecos(x1),[1,2]Let f(x)=x3. Find the second Taylor polynomial P2(x) about x0=0. Find R2(0.5) and the actual error when using P2(0.5) to approximate f(0.5). Repeat (a) with x0=1. Repeat (b) for the polynomial found in (c).6EFind the second Taylor polynomial P2(x) for the function f(x)=excosx about x0=0. Use P2(0.5) to approximate f(0.5). Find an upper bound for error f(0.5)P2(0.5) using the error formula, and compare it to the actual error. Find a bound for the error f(x)P2(x) in using P2(x) to approximate f(x) on the interval [0, 1]. Approximate 01f(x)dx using 01P2(x)dx. Find an upper bound for the error in (c) using 01R2(x)dx, and compare the bound to the actual error.8E9EUse a Taylor polynomial about /4 to approximate cos42 to an accuracy of 106.11E12E13EThe 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. Explain why this description is accurate. Find the quadratic polynomial that best approximates a function f near x0=1 if the tangent line at x0=1 has equation y=4x1, and if f(1)=6.15E16ECompute the absolute error and relative error in approximations of p by p*. p=,p*=227 p=,p*=3.1416 p=e,p*=2.718 p=2,p*=1.414 p=e10,p*=22000 p=10,p*=1400 p=8!,p*=39900 p=9!,p*=18(9/e)92EUse 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. Solve the following linear systems using four-digit rounding arithmetic. 133+0.921 1330.499 (1210.327)119 (121119)0.327 1314672e5.4 10+6e362 2997 2271174E5E6E7E8E9E10E11E12E13E1E2E3E4E5E6EUse three-digit chopping arithmetic to compute the sum i=1101/i2 first by 11+14++1100 and then by 1100+181++11. Which method is more accurate, and why?8E9E10E11E12E13EUse the Bisection method to find p3 for f(x)=xcosx on [0,1].Let f(x)=3(x+1)x12(x1). Use the Bisection method on the following intervals to find p3. [2,1.5] [1.25,2.5]Use the Bisection method to find solutions accurate to within 102 for x37x2+14x6=0 on each interval. a. [0,1] b. [1,3.2] c. [3.2,4]Use the Bisection method to find solutions accurate to within 102 for x42x34x2+4x+4=0 on each interval. [2,1] [0,2] [2,3] [1,0]Sketch the graphs of y=x and y=2sinx. Use the Bisection method to find an approximation to within 102 to the first positive value of x with x=2sinx.Sketch the graphs of y=x and y=tanx. Use the Bisection method to find an approximation to within 102 to the first positive value of x with x=tanx.Let f(x)=(x+2)(x+1)x(x1)3(x2). To which zero of f does the Bisection method converge for the following intervals? [3,2.5] [2.5,3] [1.75,1.5] [1.5,1.75]Let f(x)=(x+2)(x+1)2x(x1)3(x2). To which zero of f does the Bisection method converge for the following intervals? [-1.5,2.5] [-0.5,2.4] 10.5,31 [3,0.5]Use the Bisection method to find an approximation to 3 correct to within 104. [Hint: Consider f(x)=x23.]Use the Bisection method to find an approximation to 253 correct to within 104Find a bound for the number of Bisection method iterations needed to achieve an approximation with accuracy 103 to the solution of x3+x4=0 lying in the interval [1,4]. Find an approximation to the root with this degree of accuracy.12EThe function defined by f(x)=sinx has zeros at every integer. Show that when 1a0 and 2b3, the Bisection method converges to 0, if a+b2 2, if a+b2 1, if a+b=2Let f(x)=x26,p0=3, and p1=2. Find p3 using each method. Secant method method of False Position2EUse the Secant method to find solutions accurate to within 104 for the following problems. x32x25=0, on [1,4] x3+3x21=0, on [-3,-2] xcosx=0, on [0,/2] x0.80.2sinx=0, on [0,/2]Use the Secant method to find solutions accurate to within 105 for the following problems. 2xcos2x(x2)2=0 on [2,3] and on [3,4] (x2)2lnx=0 on [1,2] and on [e, 4] ex3x2=0 on [0,1] and on [3,5] sinxex=0 on [0,1], on [3,4] and on [6,7]5E6EUse the Secant method to find all four solutions of 4xcos(2x)(x2)2=0 in [0,8] accurate to within 105.8E9E10E11E12EThe fourth-degree polynomial f(x)=230x4+18x3+9x2221x9 has two real zeros, one in [-1,0] and the other in [0,1]. Attempt to approximate these zeros to within 106 using each method. a. method of False Position b. Secant methodThe function f(x)=tanx6 has a zero at (1/) arctan 60.447431543. Let p0=0 and p1=0.48 and use 10 iterations of each of the following methods to approximate this root. Which method is most successful and why? Bisection method method of False Position Secant methodThe sum of two numbers is 20. If each number is added to its square root, the product of the two sums is 155.55. Determine the two numbers to within 104.A trough of length L has a cross section in the shape of a semicircle with radius r. (See the accompanying figure.) When filled with water to within a distance h of the top, the volume, V of water is V=L0.5r2r2arcsinhrhr2h21/2 Suppose L=10ft,r=1ft, and V=12.4ft3. Find the depth of water in the trough to within 0.01ft.17ELet f(x)=x26 and p0=1. Use Newtons method to find p2.2E3E4E5EUse Newtons method to find all solutions of x2+10cosx=0 accurate to within 105.7E8E9E10E11E12E13E14E15E16E17E18E19E20EThe following sequences are linearly convergent. Generate the first five terms of the sequence qn using Aitkens 2 method. p0=0.5,pn=2epn1+pn12/3, for n1 p0=0.75,pn=epn1/31/2, for n1 p0=0.5,pn=3pn1, for n1 p0=0.5,pn=cospn1, for n12E3E4E5E6E7E8EFind the approximations to within 104 to all the real zeros of the following polynomials using Newtons method. P(x)=x32x25 P(x)=x3+3x21 P(x)=x3x1 P(x)=x4+2x2x3 P(x)=x3+4.001x2+4.002x+1.101 P(x)=x5x4+2x33x2+x42E3ERepeat Exercise 2 using Mullers method.5E6E7E8E9ETwo ladders crisscross an alley of width W. Each ladder reaches from the base of one wall to some point on the opposite wall. The ladders cross at a height H above the pavement. Find W given that the lengths of the ladders are x1=20ft and x2=30ft and that H=8ft. (See the figure.)11E12E1E2EUse appropriate Lagrange interpolating polynomials of degrees $1,2,$ and 3 to approximate each of the following: f(8.4) if f(8.1)=16.94410.f(8.3)=17.56492.,f(8.6)=18.50515,f(8.7)=18.82091 f13 if f(0.75)=0.07181250,f(0.5)=0.02475000,f(0.25)=0.33493750 f(0)=1.10100000 f(0.25) if f(0.1)=0.62049958,f(0.2)=0.28398668,f(0.3)=0.00660095,f(0.4)= 0.24842440 f(0.9) if f(0.6)=0.17694460,f(0.7)=0.01375227,f(0.8)=0.22363362,f(1.0)= 0.65809197Use Nevilles method to obtain the approximations for Exercise 3.5E6E7EUse the Lagrange interpolating polynomial of degree 3 or less and four-digit chopping arithmetic to approximate cos 0.750 using the following values. Find an error bound for the approximation. cos0.698=0.7661cos0.733=0.7432cos0.768=0.7193cos0.803=0.6946 The actual value of cos0.750 is 0.7317 (to four decimal places). Explain the discrepancy between the actual error and the error bound.9E10E11ENevilles method is used to approximate f(0.5), giving the following table. Determine P2=f(0.7).13ESuppose xj=j for j=0,1,2,3 and it is known that P0.1(x)=2x+1,P0.2(x)=x+1,andP1,2,3(2.5)=3 Find P0.1.23(2.5).Nevilles method is used to approximate f(0) using f(2),f(1),f(1), and f(2). Suppose f(1) was overstated by 2 and f(1) was understated by 3. Determine the error in the original calculation of the value of the interpolating polynomial to approximate f(0).16E17E1E2E3E4E5E6E7E8EA fourth-degree polynomial P(x) satisfies 4P(0)=24,3P(0)=6, and 2P(0)=0, where P(x)=P(x+1)P(x). Compute 2P(10).10EThe Newton forward divided-difference formula is used to approximate f(0.3) given the following data. x0.00.20.40.6f(x)15.021.030.051.0 Suppose it is discovered that f(0.4) was understated by 10 and f(0.6) was overstated by 5. By what amount should the approximation to f(0.3) be changed?For a function f, the Newtons interpolatory divided-difference formula gives the interpolating polynomial P3(x)=1+4x+4x(x0.25)+163x(x0.25)(x0.5) on the nodes x0=0,x1=0.25,x2=0.5 and x3=0.75. Find f(0.75).13EUse Hermite interpolation to construct an approximating polynomial for the following data.2EUse the following values and five-digit rounding arithmetic to construct the Hermite interpolating polynomial to approximate sin 0.34. Determine an error bound for the approximation in part (a) and compare to the actual error. Add sin0.33=0.32404 and cos0.33=0.94604 to the data and redo the calculations.Let f(x)=3xexe2x Approximate f(1.03) by the Hermite interpolating polynomial of degree at most 3 using x0=1 and x1=1.05. Compare the actual error to the error bound. Repeat (a) with the Hermite interpolating polynomial of degree at most 5, using x0=1, x1=1.05, and x2=1.07.5EThe following table lists data for the function described by f(x)=e0.1x2. Approximate f(1.25) by using H5(1.25) and H3(1.25), where H5 uses the nodes x0=1,x1=2, and x2=3 and H3 uses the nodes x0=1 and x1=1.5. Find error bounds for these approximations.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. Use a Hermite polynomial to predict the position of the car and its speed when t=10s Use the derivative of the Hermite polynomial to determine whether the car ever exceeds a 55-milh speed limit on the road. If so, what is the first time the car exceeds this speed? What is the predicted maximum speed for the car?8E1E2EConstruct the natural cubic spline for the following data.The data in Exercise 3 were generated using the following functions. Use the cubic splines constructed in Exercise 3 for the given value of x to approximate f(x) and f(x), and calculate the actual error. f(x)=xlnx; approximate f(8.4) and f(8.4) f(x)=sinex2; approximate f(0.9) and f(0.9) f(x)=x3+4.001x2+4.002x+1.101; approximate f13 and f13 f(x)=xcosx2x2+3x1; approximate f(0.25) and f(0.25)Construct the clamped cubic spline using the data of Exercise 3 and the fact that f(8.3)=3.116256 and f(8.6)=3.151762 f(0.8)=2.1691753 and f(1.0)=2.0466965 f(0.5)=0.7510000 and f(0)=4.0020000 f(0.1)=3.58502082 and f(0.4)=2.16529366Repeat Exercise 4 using the clamped cubic splines constructed in Exercise 5.7EConstruct a natural cubic spline to approximate f(x)=ex by using the values given by f(x) at x=0,0.25,0.75, and 1.0. Integrate the spline over [0,1], and compare the result to 01exdx=11/e Use the derivatives of the spline to approximate f(0.5) and f(0.5), and compare the approximations to the actual values.9E10E11EA clamped cubic spline s for a function f is defined on [1,3] by s(x)=s0(x)=3(x1)+2(x1)2(x1)3,if1x2s1(x)=a+b(x2)+c(x2)2+d(x2)3,if2x3 Given f(1)=f(3), find a, b, c, and d.13E14ESuppose that f(x) is a polynomial of degree 3. Show that f(x) is its own clamped cubic spline but that it cannot be its own natural cubic spline.Suppose the data xi,fxi)i=1n lie on a straight line. What can be said about the natural and clamped cubic splines for the function f? Hint: Take a cue from the results of Exercises 1 and 2.1.The data in the following table give the population of the United States for the years 1960 to 2010 and were considered in Exercise 16 of Section 3.2 and Exercise 6 of Section 3.3. Year196019701980199020002010Population(thousands)179,323203,302226,542249,633281,442307,746 Find a natural cubic spline agrecing with these data, and use the spline to predict the population in the years 1950,1975, and 2020 Compare your approximations with those previously obtained. If you had to make a choice. which interpolation procedure would you choose?18E19EIt is suspected that the high amounts of tannin in mature oak leaves inhibit the growth of the winter moth (Operophtera bromata L. Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree. Use a natural cubic spline to approximate the average weight curve for each sample. Find an approximate maximum average weight for each sample by determining the maximum of the spline.1E2EConstruct and graph the cubic BĂ©zier polynomials given the following points and guide points. Point (1,1) with guide point (1.5,1.25) to point (6,2) with guide point (7,3) Point (1,1) with guide point (1.25,1.5) to point (6,2) with guide point (5,3) Point (0,0) with guide point (0.5,0.5) to point (4,6) with entering guide point (3.5,7) and exiting guide point (4.5,5) to point (6,1) with guide point (7,2) Point (0,0) with guide point (0.5,0.25) to point (2,1) with entering guide point (3,1) and exiting guide point ( 3,1 ) to point ( 4,0 ) with entering guide point ( 5,1 ) and exiting guide point (3,-1) to point (6,-1) with guide point (6.5,-0.25)4E1E2E3E4E5E6E7E8E9E10E11E12E13E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E1E2E3E4E5E6E7E8E9E10E11E12E13E14E1E2E3E4E5E6E7E8E1E2E3E4E5E6E7E8E9E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E1E2E3E4E5E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E1E2E3E4E5E6E7E8E9E10E11E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E