The Quadratic Formula - Searching for your RootsJeremy JohnsonThe purpose of this lab is for students to begin using Maple to empirically explore mathematical concepts and to use Maple's programming constructs to automate some computations that would be laborious to do manually. The lab will guide you through some experiments which will lead you to discover some theorems which explain the behaviour of the roots of quadratic equations. Students will need to use some elementary programming constructs (loops, conditionals, lists) to carry out the experiments. This is an important step towards becoming a proficient Maple user, and more generally making effective use of a computer to automate tasks.In preparing for the lab, we will introduce some of Maple's programming constructs, review the quadratic formula and complex numbers. We will also see how Maple can be used to generate random numbers. Random number generation is necessary in order to generate inputs for our empirical investigation. We will empirically determine the probability that a random quadratic equation has two real roots. This is done by generating random quadratic polynomials and determining the percentage of them that have two real roots. In the remainder of the lab, we will systematically investigate when a quadratic polynomial has two real roots and theoretically determine the desired probability.
<Text-field style="Heading 1" layout="Heading 1">Loops and Conditionals</Text-field>Maple has a built-in programming language which can be used to extend the functionality of Maple by adding new functions and can be used to develop scripts that automate tasks that allow us to perform a large number of computations that would be tedious to enter by hand.A sequence of tasks can be performed using a loop. The basic form of a "for" loop in Maple is given in the following example. The loop iterates five times, and in the ith iteration a print statement is executed. The index variable i, is used to keep track of the iteration that is being performed and can be used as input to the computation that is performed inside the loop, so that different behaviour can occur for each iteration. The following shows the form of the for loop that was used. The text in angle brackets indicates that some other Maple code has to be inserted and the text by itself are key words that must be entered as is. <name> indicates a variable name which is used for the loop index, the first <expr> is a Maple expression which when evaluated provides the initial value for the loop index and the second <expr> is a Maple expression indicates when the loop is finished. Each iteration the loop index is incremented by one and if the value of the index is greater than the final expression the loop terminates, otherwise the sequence of Maple statements denoted by <statement sequence> are executed.for <name> from <expr> to <expr> do <statement sequence> end do; Maple provides additional types of loops, and the for loop has more general features than is illustrated by our examples; however, this is sufficient for what is needed in this lab. Additional information can be found using help.for i from 1 to 5 do print("Hello");end do;for i from 1 to 5 do print("Hello for the ",i,"th time");end do;Notice in the previous example, that "th" is printed every iteration, however, on certain iterations it would be more appropriate to print something else. For example, after the 1st iteration, it would be best to print "st" and after the second iteration it would be best to print "nd". We will see shortly that a conditional statement can be used to perform different actions depending on the value of the loop index i.At times it is necessary to instruct Maple to take different actions depending on the result of a previous calculation. Conditional statements can be used to carry out one of several possible statements depending on the result of some condition (e.g. whether a variable has a particular value, whether some expression is positive, greater than a particular value, etc.). An if statement can be used to execute a sequence of statements only if a specified condition is true. Optionally an else clause is available so that if the condition is not true an alternate sequence is executed. Finally, an optional elif clause can be included so that a list of alternate conditions are evaluated and the statement sequences following the first condition whose value is true is executed. The following shows the general form of Maple's if statement (phrases located between vertical bars are optional). Examples of all forms of the if statement are given below. if <conditional expression> then <statement sequence> | elif <conditional expression> then <statement sequence> | | else <statement sequence> | end ifFor each example that follows, the variable 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 is set to a particular value preceding the if statement that is given. Depending on the value of 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, different actions are taken. You should experiment with different valued of 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 and see what happens.i := 1;if (i = 1) then print("st");end if;i := 1;if (i = 1) then print("st");else print("th");end if;i := 2;if (i = 1) then print("st");else if (i = 2) then print("nd"); end if;end;i := 3;if (i = 1) then print("st");elif (i = 2) then print("nd");else print("rd");end if;Now, with our knowledge of if statements, it is possible to rewrite the initial loop so that the appropriate ending is printed following the loop iteration number. Before doing so, we show a more refined print statement, printf, which allows more precise formatting of output.The function printf, uses a format string to indicate text that is to be printed, the format of variables that are to be printed, and special formatting information (such as new line). In this example, an integer is printed within the string "hello for the _-th time", where "_" is to be filled in with an integer. The format specification %d indicates that an integer is to be printed. The special character "\n" indicates that a new line should be inserted after the text "time".for i from 1 to 5 do printf("hello for the %d-th time\n",i);end do;Now we can use an if statement to append different endings, "st", "nd", "rd, or "th", following the loop index 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for i from 1 to 5 do printf("hello for the %d-",i); if (i = 1) then printf("st"); elif (i = 2) then printf("nd"); elif (i = 3) then printf("rd"); else printf("th"); end if; printf(" time\n");end do;A nested loop is a loop whose body is another loop. This is necessary, for instance, if you want to generate all pairs [i,j] with 1 <= i,j <= N. For each i you want to generate all j between 1 and N. The following doubly nested loop prints all pairs [i,j], where i and j range from 1 to 5.for i from 1 to 5 do for j from 1 to 5 do print([i,j]); end do;end do;
<Text-field style="Heading 1" layout="Heading 1">List Processing</Text-field>A list is either empty, denoted by [], or contains a sequence of elements [elt1,...,eltn]. The elements of a non-empty list can be obtained using the op command. The nops command returns the number of elements in a list and can be applied to an empty or non-empty list. Two sequences of elements can be concatenated (joined together) using the comma operator. Thus an element can be added to a list, by obtaining the sequence of elements from the list, concatenating the new element in the front or back of the sequence and putting the new sequence in a list.L := [1,2,3];nops(L);op(L);L := [0,op(L)];M := [4,5,6];More general concatenation of lists is possible using the same construction.[op(L),op(M)];
<Text-field style="Heading 1" layout="Heading 1">Complex Numbers and the Quadratic Equation</Text-field>This section reviews complex numbers, the quadratic formula, and discusses the difference between the Maple functions.The Maple symbol 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 denotes 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, which is a solution to the equation 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. Once we have the symbol I, all roots of 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 can be written as expressions involving 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, I.E. 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 and 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. Note that the fundamental theorem of algebra states that once you have I, every polynomial with real coefficients has a root which can be expressed as an expression involving real numbers and the symbol 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, I.E. complex numbers. For quadratic equations, we know this from the quadratic formula.Complex numbers are of the form 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, where 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 and 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 are real numbers. The real number 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 is called the real part of the complex number 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 and the number 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 is called the imaginary part of 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. Two complex numbers can be multiplied using the distributive and commutative properties along with the fact that 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. 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The inverse of a complex number z is a complex number w such that 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. Such a number w can be found by observing that a complex number 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 multiplied by its complex conjugate 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 is equal to the real number 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: 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. Therefore, the inverse of 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 is equal to 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. Division by a complex number is performed by multiplying by the inverse.solve(x^2+1);I^2;subs(x=I,x^2+1);z := 1+I;w := 2-I;Re(w);Im(w);conjugate(w);z*w;w*conjugate(w);The conjugate of the product of two complex numbers is equal to the product of the conjugates of the two numbers. We can verify this for a particular case using Maple. However, we can also use Maple to prove this, by checking the property for general complex numbers.conjugate(z*w) = conjugate(z)*conjugate(w);Let 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 and 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, where 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, 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, 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, and 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 are parameters representing arbitrary real numbers. Maple's assume facility is used to indicate that the parameters a, b, c, a d are real. After making the assumptions, the variables are displayed with a tilde (~) to remind users that the variables have an assumption associated with them.assume(a,real); assume(b,real); assume(c,real); assume(d,real);z := a+b*I; w := c+d*I;z*w;p := expand(z*w);Re(p);Im(p);Now let's prove that 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, where bar indicates conjugation. 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 if and only if 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.conjugate(z*w) - conjugate(z)*conjugate(w);This property can be used to prove that if the roots of a real equation are complex, they must come in complex conjugate pairs. For example, if 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, and 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 and 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, then 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. If the coefficients of 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 are real, then 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 and 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, which is equal to 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. Thus LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2JkYrLUkmbW92ZXJHRiQ2JS1GLDY5USZhbHBoYUYnRi9GMkY1L0Y5RjdGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW4vRmhuUSdub3JtYWxGJ0Zqbi1JI21vR0YkNjNRLSZjb25qdWdhdGUwO0YnLyUlZm9ybUdRJmluZml4RicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZkcC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobWF4c2l6ZUdRKWluZmluaXR5RicvJShtaW5zaXplR1EiMUYnLyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJTBmb250X3N0eWxlX25hbWVHRlYvJSVzaXplR0Y0LyUrZm9yZWdyb3VuZEdGQy8lK2JhY2tncm91bmRHRkNGZXEtRmhvNjNRIj1GJ0ZbcEZecEZgcC9GY3BRL3RoaWNrbWF0aHNwYWNlRicvRmZwRmNyRmdwRmlwRltxRl5xRmFxRmNxRmVxRmdxRmlxRltyRl1yLUYsNjlRJWJldGFGJ0YvRjJGNUZkb0Y7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZlb0ZqbkYr and 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. This observation generalizes to polynomials of arbitary degree. For quadratic polynomials, this property also follows from the quadratic formula.A := a*x^2+b*x+c;solve(A,x);This formula implies that if 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 is negative that the two roots are complex numbers and come in complex conjugate pairs.A := x^2+3*x+5;3^2-4*1*5;solve(A);
<Text-field style="Heading 1" layout="Heading 1">Random Numbers</Text-field>Mathematical properties can be investigated empirically be generating random inputs and testing the property on the random sample. From this data it may be possible to conjecture theorems which can be further investigated and perhaps proven.The Maple function rand(a..b), produces a random number generator that generates numbers in the range a..b with a uniform distribution (i.e. any number in the range is equally likely to occur).die := rand(1..6);seq(die(),i=1..10);Let's generate a largern number of samples and check the distribution. Since each number between 1 and 6 is equally likely to occur, we would expect that the number of times each number occurs to be roughly (not exactly) equal. We can check this visually using the histogram function.S := [seq(die(),i=1..1000)]:with(stats[statplots]);histogram(S,area=count,numbars=6);Let's try an even bigger sample. We can do really large experiments (quite quickly) with the computer. Don't be afraid to perform large experiments, but make sure that you do a small experiment first to make sure that everything is working properly. Also think about what you are doing (what do you expect to happen) before doing the experiment. You should have some clear hypothesis in mind so that the data you collect can be used to confirm, support or refute your hypothesis. If you do not have a hypothesis to check, you should atleast have in mind the question you are investigating and the possible outcomes so that you can look at the data produced from the experiment in an intelligent manner. In this case, we simply expect the number of occurrence of each number to occur roughly the same number of times.In this experiment, the list of samples is created in a different fashion (I did not use seq), where a single sample is prepended to a list, which is initialized to zero. This approach is no better than using seq, however, in some cases seq can not be easily used and you should be aware of this approach.S := []; NumRolls := 10000;for i from 1 to NumRolls do S := [die(),op(S)]:end do:nops(S);histogram(S,area=count,numbars=6);
<Text-field style="Heading 1" layout="Heading 1">Question 1: Loops</Text-field>In the introduction to loops it was shown how to write a nested loop to generate all pairs [i,j]. with 1 <= i, j <= 5. In this question you are to modify the code so that only pairs with i < j are printed.for i from 1 to 5 do for j from i+1 to 5 do print([i,j]); end do;end do;
<Text-field style="Heading 1" layout="Heading 1">Question 2: Random Quadratic Equations</Text-field>This part of the lab empirically determines the probability that a random quadratic equation has real roots. You can use the Maple function rand to generate random quadratic polynomials (Maple also has a function called randpoly, but we will use rand).We will generate random quadratic polynomials of the form 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2OVEiYkYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdGQy8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNy8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkMvJStmb250ZmFtaWx5R0YxLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lKW1hdGhzaXplR0Y0 and 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 are uniformly distributed between -99 and 99. After generating a random polynomial, you can use solve to find its roots and can use the type command to check to see if the roots are both real. solve returns a sequence of roots, and you can use the index operator [] to access the first and second roots in the sequences.restart;randcoef := rand(-99..99):A := x^2 + randcoef()*x + randcoef();solve(A);r1 := %[1]; r2 := %%[2];type(r1,realcons);Now you are ready to compute a random sample of quadratic equations and empirically determine the probability that the equations of real roots.In order to do this, you need to write a loop which iterates over the number of samples you will generate. For each iteration, generate a random quadratic equation and then use solve and type to check to see if the roots are real. You should keep a count of the number of equations that have real roots. When all samples have been generated, compute the percentage that have real roots.Do this for 100, 1000, and 10000 samples. You should use a variable (e.g. NumPoly and RealCount) to specify the number of polynomials to generate and the number which have real roots - i.e. don't hard code constants. What conclusion do you draw from your three experiments?You should estimate how much time your experiments (or atleast the two larger experiments) will take before running the experiments so you have some idea of how much time it will take. You can estimate the time to do the experiment for 1000 polynomials once you know the time for 100 (how?). Similarly you can estimate for 10000 samples.NumPoly := 100; RealCount := 0;for i from 1 to NumPoly do A := x^2 + randcoef()*x + randcoef(); r := solve(A): if type(r[1],realcons) then RealCount := RealCount + 1: end if:end do:evalf(RealCount/NumPoly);NumPoly := 1000; RealCount := 0;for i from 1 to NumPoly do A := x^2 + randcoef()*x + randcoef(); r := solve(A): if type(r[1],realcons) then RealCount := RealCount + 1: end if:end do:evalf(RealCount/NumPoly);NumPoly := 10000; RealCount := 0;for i from 1 to NumPoly do A := x^2 + randcoef()*x + randcoef(); r := solve(A): if type(r[1],realcons) then RealCount := RealCount + 1: end if:end do:evalf(RealCount/NumPoly);
<Text-field style="Heading 1" layout="Heading 1">Question 3: When a Quadratic has Real Roots</Text-field>In this question, you will further investigate the probability that a random quadratic equation has real roots. This investigation should shed some light on the results obtained previously. Recall the quadratic equation.restart;with(plots):A := a*x^2+b*x+c;solve(A,x);This implies that a quadracit equation has real roots if and only if 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. We will systematically generate all quadratic equations with integer coefficients less than a specified size and will plot the coefficients [a,b,c] of those equations that have real roots with red points and those that have complex (non-real) roots blue . The percentage of red points is equal to the probability that random quadratic equations, whose coefficients are integers uniformly distributed in a specified range, have real roots. Since, we can always divide by the leading coefficient, i.e. a, we will assume that a is 1. This allows us to have a two dimensional plot, when plotting the points [b,c]. This is consistent with the experiment performed in question 2.Write a nested loop that generates all quadratic equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSptYXRoY29sb3JHRkMvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRjEvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUpbWF0aHNpemVHRjQtRiM2KEYrLUYjNiVGKy1JJW1zdXBHRiQ2JS1GLDY5USJ4RidGL0YyRjVGOEY7Rj1GP0ZBRkRGRkZIRkpGTEZORlBGUkZURldGWUZlbkZnbkZqbi1JI21uR0YkNjlRIjJGJ0YvRjJGNS9GOUY3RjtGPUY/RkFGREZGRkhGSkZMRk5GUEZSRlRGV0ZZRmVuL0ZoblEnbm9ybWFsRidGam4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRistSSNtb0dGJDYzUSIrRicvJSVmb3JtR1EmaW5maXhGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUnbHNwYWNlR1EwbWVkaXVtbWF0aHNwYWNlRicvJSdyc3BhY2VHRl1xLyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShtYXhzaXplR1EpaW5maW5pdHlGJy8lKG1pbnNpemVHUSIxRicvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lMGZvbnRfc3R5bGVfbmFtZUdGVi8lJXNpemVHRjQvJStmb3JlZ3JvdW5kR0ZDLyUrYmFja2dyb3VuZEdGQy1GIzYlLUYsNjlRImJGJ0YvRjJGNUY4RjtGPUY/RkFGREZGRkhGSkZMRk5GUEZSRlRGV0ZZRmVuRmduRmpuLUZhcDYzUTEmSW52aXNpYmxlVGltZXM7RidGZHBGZ3BGaXAvRlxxUSQwZW1GJy9GX3FGYXNGYHFGYnFGZHFGZ3FGanFGXHJGXnJGYHJGYnJGZHJGZnJGY29GYHAtRiw2OVEiY0YnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam5GKw== and store all points [b,c] such that 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 has real roots in a list called RealCoefs and those points [b,c] such that 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 does not have real roots in a list called ImagCoeffs. You may use the condition 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 to determine if the equation has real roots. You should initialize the lists RealCoeffs and ImagCoeffs to empty lists and add the pairs [b,c] to the appropriate list. Determine the percentage of equations that have real roots (you can either count in the loops as before, or simply use nops to determine the lengths of the lists when the loops finish executing.size := 5;RealCount := 0; RealCoefs := []; ImagCoefs := [];for b from -size to size do for c from -size to size do A := x^2 + b*x + c; if b^2 - 4*c >= 0 then RealCount := RealCount + 1; RealCoefs := [[b,c],op(RealCoefs)]; else ImagCoefs := [[b,c],op(ImagCoefs)]; end if; end do;end do;evalf(RealCount/(2*size +1)^2);evalf(nops(RealCoefs)/(nops(RealCoefs)+nops(ImagCoefs)));Produce a plot (don't forget that you can assign plots to variables and can use display to simultaneously display multiple plots) that shows all points 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 that correspond to equations 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 that have two real roots as red and all points 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 that corresponds to equations 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 with complex roots that are not real.Include in your plot the curve 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. What information does this curve give?plot1 := pointplot(RealCoefs,color=red):plot2 := pointplot(ImagCoefs,symbol=DIAMOND,color=blue):b := 'b';plot3 := plot(b^2/4,b=-10..10):display({plot1,plot2,plot3});Do the experiment again with size = 10. How does the probabilty compare to that with size 5? Draw the plot as before, but this time with the data for size 10.size := 10;RealCount := 0; RealCoefs := []; ImagCoefs := [];for b from -size to size do for c from -size to size do A := x^2 + b*x + c; if b^2-4*c>=0 then RealCount := RealCount + 1; RealCoefs := [[b,c],op(RealCoefs)]; else ImagCoefs := [[b,c],op(ImagCoefs)]; end if; end do;end do;evalf(RealCount/(2*size +1)^2);evalf(nops(RealCoefs)/(nops(RealCoefs)+nops(ImagCoefs)));plot1 := pointplot(RealCoefs,color=red):plot2 := pointplot(ImagCoefs,symbol=DIAMOND,color=blue):b := 'b';plot3 := plot(b^2/4,b=-10..10):display({plot1,plot2,plot3});Do the experiment one more time with size = 20. How does the probabilty compare to that with sizes 5 and 10? Draw the plot as before, but this time with the data for size 20.size := 20;RealCount := 0; RealCoefs := []; ImagCoefs := [];for b from -size to size do for c from -size to size do A := x^2 + b*x + c; if b^2-4*c>=0 then RealCount := RealCount + 1; RealCoefs := [[b,c],op(RealCoefs)]; else ImagCoefs := [[b,c],op(ImagCoefs)]; end if; end do;end do;evalf(RealCount/(2*size +1)^2);evalf(nops(RealCoefs)/(nops(RealCoefs)+nops(ImagCoefs)));plot1 := pointplot(RealCoefs,color=red):plot2 := pointplot(ImagCoefs,symbol=DIAMOND,color=blue):b := 'b';plot3 := plot(b^2/4,b=-20..20):display({plot1,plot2,plot3});
<Text-field style="Heading 1" layout="Heading 1">Bonus Question: Question 3 in the Limit</Text-field>In the previous question you investigated the probability that 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 had two real roots where b and c are in the range from -N to N for N = 5, 10, and 20. In this bonus question, you will investigate what happens to the probability when N goes to infinity.In order to do this, you will estimate the area of the square whose corners are [-N,-N], [N,-N], [N,N], [-N,N]where 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 is negative. In order to compute this area, you will need to use the int command, which can be used to compute the area under a curve. "int" stands for integration and you will learn all about this in the second term of your calculus course. For now it suffices to think of int as the area under a curve. For example, the area under the curve y = x as x goes from 0 to 1 is equal toint(x,x=0..1);plot(x,x=0..1);You already know how to compute this area using the formula for the area of a triangle; however, int can be used for more complicated curves. For example, the following command shows the area under the curve x^2 as x goes from 0 to 1.int(x^2,x=0..1);Note that the area is less than the area under the curve y=x as can be seen from the following plot.plot1 := plot(x,x=0..1):plot2 := plot(x^2,x=0..1):display({plot1,plot2});