Exam (elaborations) TEST BANK FOR Mathematics for Physical Science and

Exam (elaborations) TEST BANK FOR Mathematics for Physical Science and The scientific world is presently in a transitional state with regard to symbolic computing, and there are still many physical scientists (and even engineers) who are not yet fluent in any symbolic computing language. This population even includes members of the instructional faculties of distinguished educational institutions. But despite that situation, it is important to include the use of symbolic computing tools in the education of the next generation of scientists, engineers, and their teachers. Fortunately, it is not necessary that the instructor in a course using this text be truly expert in symbolic computation. But it is of course assumed that he/she have an appropriate background in the mathematics the text is designed to teach. To assist instructors in preparing for the tasks they face, this manual provides, in addition to problem answers, complete coding for Exercises that are to be completed using symbolic computation. An important decision that will need to be made before using this text is that of the computer language to be used. One could leave this choice to individual students or present both languages, but the instructor may wish to consider issues such as the language availability and cost, and in addition the preparative effort needed. An instructor who uses maple should become familiar with its help facility (use the command ?topic). The maple help pages are rather complete, and, though sometimes opaque to students, can provide answers to vexing problems. It is also important to learn the editing features needed to make corrections to Worksheet input, namely the split and join options of the edit pull-down menu. An attempt to use Enter to open space in previously-written code is interpreted by maple as a command to execute it, so the editing techniques are important to avoid that action. Without good command of the editing features it is difficult to build easily understood code. Instructors who use mathematica might want to obtain (from its free on-line documentation) the two books Core Language and Mathematics and Algorithms. These sources contain examples of the use of almost all the features relevant for the present book. There is no special technique needed to correct errors or make insertions in Notebooks. Because mathematica uses Shift-Enter as its command-termination signal, multi-line insertions can just be typed where they are needed. 1 2 CHAPTER 0. INTRODUCTION GETTING STUDENTS ON-LINE The text contains no information describing how students can initiate a symbolic computing session or obtain stored or printed copies of their work. The instructor will need to provide, externally to the text, all the information that is needed in that area. The ideal situation is for students to have access to symbolic computing using their own computers and with an appropriate group license. An alternative can be access through an institution’s computer lab, but in that case it is important that the access hours and availability be adequate. Chapter 1 COMPUTERS, SCIENCE, AND ENGINEERING 1.1 COMPUTING: HISTORICAL NOTE 1.2 BASICS OF SYMBOLIC COMPUTING Exercises Carry out these exercises using the symbolic computation system of your choice. 1.2.1. Reduce the following polynomial in x and z to fully factored form. 3x 2 + 3x 4 − 4xz + 3x 2 z + 2x 3 z + 6x 5 z + z 2 − 4xz2 − 7x 2 z 2 + 6x 3 z 2 − 8x 4 z 2 + z 3 + 2xz3 − 5x 2 z 3 + 2x 3 z 3 − 2xz4 + 6x 3 z 4 + z 5 − 8x 2 z 5 + 2xz6 Solution: In maple, > ZZ := 3*x^2+3*x^4-4*x*z+3*x^2*z+2*x^3*z+6*x^5*z+z^2-4*x*z^2 > -7*x^2*z^2+6*x^3*z^2-8*x^4*z^2+z^3+2*x*z^3-5*x^2*z^3 > +2*x^3*z^3-2*x*z^4+6*x^3*z^4+z^5-8*x^2*z^5+2*x*z^6; > factor(ZZ): In mathematica, ZZ = 3*x^2+3*x^4-4*x*z+3*x^2*z+2*x^3*z+6*x^5*z+z^2-4*x*z^2 -7*x^2*z^2+6*x^3*z^2-8*x^4*z^2+z^3+2*x*z^3-5*x^2*z^3 +2*x^3*z^3-2*x*z^4+6*x^3*z^4+z^5-8*x^2*z^5+2*x*z^6: Factor[ZZ]; Both symbolic systems give an answer equivalent to (x − z)(3x − z)(2xz + 1)(x 2 + z 3 + z + 1) 1.2.2. Because the polynomial in Exercise 1.2.1 can be factored, symbolic computer systems can easily find the values of x that are its roots. Calling the polynomial in its original form poly, use one of the following: 3 4 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING solve(poly=0,x) or Solve[poly==0,x] Note. Less convenient forms than that found here are usually obtained as the solutions to more general root-finding problems. Solution: Letting ZZ be as defined in the solution to Exercise 1.2.1, execute one of > solve(ZZ = 0, x); 1 3 z, − 1 2z , z , √ −z 3 − z − 1 , − √ −z 3 − z − 1 Solve[ZZ == 0, x] { { x → − 1 2z } , { x → z 3 } , {x → z}, { x → −√ −1 − z − z 3 } , { x → √ −1 − z − z 3 } } 1.2.3. (a) Starting with f(x) = 1 + 2x + 5x 2 − 3x 3 , obtain an expansion of f(x) in powers of x + 1 by carrying out the following steps: (1) Define a variable s and assign to it the value of the polynomial representing f(x); (2) Define x to have the value z − 1 and recompute s; (3) Expand s in powers of z; (4) Define z to have the value x + 1 and recompute s. You will need to be careful to have x and/or z undefined at various points in this process. (b) Expand your final expression for s and verify that it is correct. Note. Your algebra system may combine the linear and constant terms in a way that causes (x + 1) not to be explicitly visible. Solution: In maple, these steps correspond to this coding: > s := 1+2*x+5*x^2-3*x^3: > x:=z-1: s: > ss:=expand(s): > x:=’x’: z:=x+1: ss: In mathematica, s = 1 + 2*x + 5*x^2 - 3*x^3; x = z - 1; s; ss = Expand[s]; Clear[x]; z = x + 1; ss; Both symbolic systems give a result equivalent to 7 − 17(x + 1) + 14(x + 1)2 − 3(x + 1)3 1.2.4. Verify the trigonometric identity sin(x + y) = sin x cos y + cos x sin y by simplifying sin(x + y) − sin x cos y − cos x sin y to the final result zero. Use the commands for simplifying and expanding (try both even if the first works). These two commands do not always give the same results. Solution: Both work in maple, Expand does not simplify in mathematica. 1.2. BASICS OF SYMBOLIC COMPUTING 5 1.2.5. Obtain numerical values, precise to 30 decimal digits, for π 2 and sin 0.1π. Note. Obtain these 30-digit results even if your computer does not normally compute at this precision. Solution: Execute one of the following (illustrated only for sin 0.1π): evalf(sin(0.1*Pi), 30); N[Sin[0.1*Pi], 30] Both give sin 0.1π = 0. . The result for π 2 is 9. . 1.2.6. (a) Verify that your computer system knows the special function corresponding to the following integral: erf(x) = 2 √ π ∫ x 0 e −t 2 dt . In maple, it is erf(x); in mathematica, Erf[x]. (b) Write symbolic code to evaluate the integral defining erf(x), and check your code by comparing its output with calls to the function in your symbolic algebra system. Check values for x = 0, x = 1, and x = ∞. Note. Infinity is infinity in maple, Infinity in mathematica. Solution: erf(0) = 0, erf(1) = 0., erf(∞) = 1. evalf(2/sqrt(Pi)*int(exp(-t^2),t=0 .. 1)); N[2/Sqrt[Pi]*Integrate[E^(-t^2), {t, 0, 1}]] 1.2.7. Plot the function defined in Exercise 1.2.6 for various ranges of x. Use enough ranges to get some experience in using your algebra system’s plotting utility. Solution: We illustrate for the range (0 · · · 2): plot(erf(x), x = 0 .. 2); Plot[Erf[x], {x, 0, 2}] 1.2.8. Obtain the value of erf(π) to 30 decimal places and print out the message "erf(pi)= · · · ". Note. The value of erf must be supplied directly by the computer; you should not obtain an answer by typing it in. Solution: In maple: VAL := evalf(erf(Pi),30): print("erf(pi) = ",VAL); “erf(pi) = ”, 0.781 or 6 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING printf("erf(pi) = %33.30fn",VAL); erf(pi)= 0.781 In mathematica, VAL = N[Erf[Pi], 30]; Print["erf(pi) = ", VAL] erf(pi)= 0.781 1.2.9. Obtain the sixth derivative with respect to x of 1/ √ x2 + z 2. Evaluate it numerically for x = 0, z = 1. Solution: maple code for this problem can be > d6 := diff(1/sqrt(x^2+z^2),x,x,x,x,x,x); In mathematica, d6 = D[1/Sqrt[x^2+z^2],x,x,x,x,x,x] Both symbolic programs give results equivalent to 10395x 6 (x 2 + z 2) 13/2 − 14175x 4 (x 2 + z 2) 11/2 + 4725x 2 (x 2 + z 2) 9/2 − 225 (x 2 + z 2) 7/2 At x = 0, z = 1 this expression evaluates to −225. To obtain this result symbolically, execute one of > x:=0: z:=1: d6; x=0; z=1; d6 1.2.10. (a) Evaluate the indefinite integral ∫ dz (z 2 + 4)7/2 . (b) Evaluate the above integral numerically for the range (0, π). Solution: (a) Use either of the following coding statements; both give essentially the same result: int(1/(z^2+4)^(7/2),z); Integrate[1/(z^2 + 4)^(7/2), z] z(30 + 10z 2 + z 4 ) 120(4 + z 2) 5/2 (b) Use one of int(1/(z^2+4)^(7/2),z = 0 .. Pi); Integrate[1/(z^2 + 4)^(7/2), {z,0,Pi}] Either of these yields 0. · · · . 1.2. BASICS OF SYMBOLIC COMPUTING 7 1.2.11. The amplitude A of the light of wavelength λ that is transmitted through a slit of width d is in the approximation of Fraunhofer diffraction given as A = A0 sin x/x, where x = πd sin θ/λ, with θ the angle from the normal to the slit; see Fig. 1.2. (a) Plot A over a range of x that is symmetric about x = 0 and contains 11 extrema. (b) The places where A = 0 occur are at x = ±π, ±2π, · · · , and the extrema lie between (but not exactly half-way between) these zeros. Using the plot facilities of your symbolic computing system, change the range of your plot in ways that permit each extremum to be located to an accuracy of 0.001 in x. (c) Returning now to the formula for A, compute the integral of A2 for each region bounded by adjacent zeros of A, and thereby determine the ratio of the intensity for the region containing x = 0 (referred to as the principal maximum) to the intensity in one of the immediately adjacent regions. Solution: (a) Use one of the following statements: plot(sin(x)/x, x=-6*Pi .. 6*Pi); Plot[Sin[x]/x, {x, -6*Pi, 6*Pi}] (b) The extrema are at x = 0, ±4.493, ±7.725, ±10.904, ±14.066, ±17.221. Illustrating for the extremum near 4.5, we can change the plot range repeatedly until reaching something similar to one of the following: plot(sin(x)/x, x = 4.493 .. 4.494); Plot[Sin[x]/x, {x, 4.493, 4.494}] (c) The intensity of the principal maximum is given by one of evalf(int(sin(x)^2/x^2, x=-Pi .. Pi)); N[Integrate[Sin[x]^2/x^2, {x,-Pi,Pi}]] Either of these yields 2.836. The intensities of other regions is given (for nonzero n) by one of evalf(int(sin(x)^2/x^2, n*x= Pi .. (n+1)*Pi)); N[Integrate[Sin[x]^2/x^2, {x, n*Pi, (n+1)*Pi}]] For successive n values these integrals have the values 0.074, 0.026, 0.013, 0.008, 0.005. The intensity of the principal maximum is 2.836/0.074 ≈ 38 times that of either adjacent region. 1.2.12. Consider the function f(x) = x 6 − 2x 5 + x 4 − 1.783x 3 + 1.234x 2 + 0.877x − 0.434. Find all the roots of f(x) to three decimal places by making plot(s) over a suitable range or ranges. Solution: The roots are most easily found by executing, in maple > f:=x^6-2*x^5+x^4-1.783*x^3+1.234*x^2+0.877*x-0.434; > solve(f); or, in mathematica, f=x^6-2*x^5+x^4-1.783*x^3+1.234*x^2+0.877*x-0.434 Solve[f == 0,x] 8 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING There are real roots near −0.5566, 0.3913, 0.9416, and 1.7040. There is a pair of complex roots near −0.2402 ± 1.0882i. The real roots can also be found graphically by making plots limited to the vicinity of individual roots. One way to find the complex roots graphically starts by defining a function f1(x + iy) obtained by dividing f(x + iy) by the product of x − xn for the real roots. We can then minimize |f1| graphically with respect to x (for fixed y) and then with respect to y (for fixed x), iterating until |f1| has been made acceptably small. 1.2.13. Given the function f(x) = a 3x 3 − x 2 + 3x − 4a, find by making suitable plots a value of a such that f(2) = 0. Is your value of a unique? Solution: Setting x = 2, we require 8a 3 − 4a + 2 = 0. The plot shows that there is only one crossing of the x-axis, so there is one real value of a and two conjugate complex values. Refining the plot range, the real value of a is approximately −0.8846. 1.3 SYMBOLIC COMPUTATION PROGRAMS Exercises Carry out these exercises using the symbolic computation system of your choice. 1.3.1. Code the problem of Example 1.3.2 and execute it for a variety of values of m and x. Verify that your code performs as expected. Solution: Run the codes given in the Example. 1.3.2. Write code that examines the values of two numerical quantities x and y, and produces as output one of the strings Case I or Case II, whichever applies. Case I is when x + y > 10 and 0 < xy < 3; Case II occurs otherwise. Solution: In maple (can test with various values of x and y): x := 20: y := 0.1: if (x+y > 10 and x*y > 0 and x*y < 3) then print("Case I") else print("Case II") end if; ”Case I” In mathematica (can test with various values of x and y): x = 20; y = -0.1; If[x + y > 10 && x*y > 0 && x*y < 3, Print["Case I"], Print["Case II"]] Case II 1.3.3. Code the first two problems of Example 1.3.3 and execute them with a variety of input parameters (n and t). For the first problem, verify that your code yields the exact values given by the formula that was presented as its solution. For the second problem, see how rapidly the finite summations converge toward the analytical result. Note. The fact that individual terms get small is not a guarantee that a series converges. In 1.3. SYMBOLIC COMPUTATION PROGRAMS 9 the present case, the series is convergent. Solution: Run the codes given in the Example. 1.3.4. Determine what happens if, in the second problem of Example 1.3.3, the decimal point is removed from the 1./n4 in the fourth line of the coding. Explain the reason for what you observe. Solution: The result is then expressed as a fraction with the numerator and denominator both large integers. 1.3.5. Write code which adds members of the series 1 + 1 2 + 1 3 + 1 4 + · · · until the sum has reached a value exceeding 10, then giving as output the number of addends used. Solution: In maple: s:=0.: n:= 1: while (s<10) do s:=s+1/n; n:=n+1 end do: n; 12368 In mathematica: s = 0.; n = 1; While[s < 10, s = s + 1/n; n = n + 1]; n 12368 The sum of this large number of terms is 10.00004 · · · . Note that s was defined as a decimal zero to avoid the inefficiency of processing fractions with extremely large numerators and denominators. 1.3.6. Write code that, by testing successive integers, finds the largest integer whose square root is less than or equal to an input number n: (a) Using the construction while or While, (b) Using the command break or Break. Solution: Using maple, test case for n = 8.75: (a) n := 8.75: j := 0: while (sqrt(1.*j) <=n) do j := j+1 end do: j-1; 76 (b) for j from 0 to infinity do if (sqrt(1.*j) >= n) then break end if; end do: j-1; 76 Using mathematica, test case for n = 7.1: (a) n = 7.1; j = 0; While[Sqrt[1.*j] <= n, j = j + 1]; j - 1 50 (b) Do[ If[Sqrt[1.*j] >= n, Break[]], {j, 0}]; j - 1 50 In both languages the index is incremented as the last operation of each iteration of a do loop. 10 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING 1.3.7. Given the definition f(n) = sin(0.27n), write code using the command next or Continue that will determine the number of integer n values in the range [1, 1000] for which |f(n)| > 0.4. Solution: In maple: k:=0: for n from 1 to 1000 do if (abs(sin(0.27*n)) <= 0.4) then next end if; k := k+1 end do: k; In mathematica: k = 0; Do[ If[Abs[Sin[0.27*n]] <= 0.4, Continue[]]; k = k + 1, {n, 1, 1000}]; k Both codes yield the result 737. 1.4 PROCEDURES Exercises Carry out these exercises using the symbolic computation system of your choice. 1.4.1. Enter the code for the procedure dfac of Example 1.4.3 and verify that it gives correct results for all integer n from −1 to 8. Solution: Run the code given in the Example. Values to check against are given in the solution to Exercise 1.5.2. 1.4.2. Convert the second problem of Example 1.3.3 into a procedure and test its application for several precision levels t. Note. This summation is ζ(4) = 1.08232. Solution: A maple procedure: Inv4powerSum := proc(t) local S,n; S := 0: n := 1: while (1/n^4) >= t do S := S + 1./n^4; n := n + 1 end do: print("Largest n used",n-1): S; end proc; Calling this procedure: Inv4powerSum(0.0001); ”Largest n used”, 10 1. 1.4. PROCEDURES 11 A mathematica procedure: inv4powerSum[t_] := Module[ {n, s}, s = 0; n = 1; While[(1/n^4) >= t, s = s + 1./n^4; n = n + 1]; Print["Largest n used = ", n - 1]; s] Calling this procedure: inv4powerSum[.0001] Largest n used = 10 1.08204 1.4.3. Enter the code for the procedure legenP of Example 1.4.5. (a) Verify that it gives correct results by checkng that P4(x) = 35 8 x 4 − 15 4 x 2 + 3 8 . and that (for several n, some even and some odd), Pn(1) = 1. (b) Remove the simplifying command expand or Expand and notice how the results are changed. Solution: (a) Run P4:=legenP(4,x) or P4=legenP[4,x], obtaining P4 = 35x 4/8 − 15x 2/4 + 3/8. (b) Without expand or Expand the recursion produces results that need simplification. 1.4.4. To see how the use of recursion in procedure definitions can make the coding cleaner, consider a procedure that computes the Legendre polynomial Pn(x) without using the recursive strategy illustrated in Example 1.4.5. Referring to that example for definitions, write a procedure altLegP that computes the Legendre polynomial Pn(x) in the following way: 1. Letting P be the polynomial Pj in Step j, and assuming that Q and R contain the polynomials of the two previous steps, write for P the formula P = [(2j − 1)xQ − (j − 1)R]/j. 2. After making P, update Q and R by moving the contents of Q into R and those of P into Q. 3. You will need starting values of Q and R to initiate this process and must organize the coding to deal with special cases and control the number of steps to be taken. Solution: In maple, altLegP := proc(n,x) local P,Q,R,j; P := 1; if (n > 0) then Q := P; P := x end if; for j from 2 to n do R := Q; Q := P; P := expand( ((2*j-1)*x*Q - (j-1)*R)/j ) end do; P end proc: 12 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING In mathematica, altLegP[n_, x_] := Module[ {p, q, r, j}, p = 1; If[n > 0, q = p; p = x]; Do[ r = q; q = p; p = Expand[((2*j-1)*x*q - (j-1)*r)/j], {j,2,n} ]; p ] 1.4.5. Generate formulas that give sin nx and cos nx in terms of sin x and cos x by defining procedures based on the formulas sin(n + 1)x = sin nx cos x + cos n