December 18, 2017 Section W Nandyalam

Size: px

Start display at page:

Download "December 18, 2017 Section W Nandyalam"

Theresa Barker
5 years ago
Views:

1 Mathematica Gift #11 Problem Complete the following 15 exercises in a Mathematica notebook. You should write the problem in a text cell and show your work below the question. Questions Question 1 Compute the sum of the reciprocals of 3, 5, 7, 9,, 63. First, determine the number of odd numbers between 3 and 63 inclusive. To do so, take the first value of the series (3) and the last number of the series (63). Subtract the two values. Divide by, and add 1 for inclusiveness = / = = 31 There are 31 odd numbers between 3 and 63. totalsum = 0; i = 1; While i 31, totalsum = totalsum + 1 * i + 1 ; i++ ; Print[N[totalSum]] The sum of the reciprocals of 3, 5, 7, 9,..., 63 is Question Page 1 of 7

2 Compute The result of this computation is Question 3 Obtain a 50 significant digit approximation to π. N π, Above is the approximation of π to the 50th significant digit. Question 4 Use the Table function to make a list of the 30 prime numbers starting with 11 and ending with 139. Name the list P. P = Table[Prime[n + 4], {n, 30}] {11, 13, 17, 19, 3, 9, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 17, 131, 137, 139} Question 5 Add the numbers in the sequence P found in question 4. Total[P] 110 Question 6 Multiply the numbers in the list P found in question 4. Then get a count of the number of digits in the product. Page of 7

3 prodp = Apply[Times, P] IntegerLength[prodP] 53 Question 7 Sketch the graphs of y = sin [x], y = sin [x], and y = sin [3x], 0 x π in steps of π/, on one set of axes. Use different colors for each curve. plotting = Plot[ {Sin[x], Sin[ x], Sin[3 x]}, {x, 0, * Pi}, PlotLegends "Expressions", Ticks {{0, Pi, Pi / }}]; Show[plotting, AxesLabel {HoldForm[x], HoldForm[y]}, y PlotLabel None, LabelStyle {GrayLevel[0]}] sin(x) π π x sin( x) sin(3 x) Question 8 Make a list of the cubes of the integers, 5, 6, 9, 1, 44. Add the numbers in the list of cubes and then display the prime factorization of the sum. What do you notice about the prime factorization? first = ^3; second = 5^3; third = 6^3; fourth = 9^3; fifth = 1^3; sixth = 44^3; Page 3 of 7

4 List[{first, second, third, fourth, fifth, sixth}] {{8, 15, 16, 79, 178, }} eightsum = first + second + third + fourth + fifth + sixth FactorInteger[eightSum] {{, 1}, {3, 1}, {5, 1}, {7, 1}, {419, 1}} I noticed that through the prime factorization, all the exponents are 1. Question 9 Obtain the prime factorization of the product of the integers in the list of cubes described in Exercise 8. eightproduct = first * second * third * fourth * fifth * sixth FactorInteger[eightProduct] {{, 18}, {3, 1}, {5, 3}, {11, 3}} Question 10 Find two ways to find an approximate value of x for which ^x = 100. Display the solution in a graph. sol1 = NSolve[^x == 100, x, Reals] {{x }} sol = Solve[Log[, 100.] x, x] {{x }} xvalue = ` Page 4 of 7

5 o = ListPlot[{xvalUE}, PlotLabel None, PlotStyle {Red, PointSize[0.03]}] dissol = Plot[^x == 100, {x, 0, 100}]; Show[{disSol, o}, AxesLabel {HoldForm[x], HoldForm[y]}, PlotLabel None, LabelStyle {GrayLevel[0]}] y x Question 11 What is the 115th Fibonacci number? What is the 1,115th Fibonacci number? Fibonacci[115] Fibonacci[1115] % % % Page 5 of 7

6 Question 1 What are the greatest common divisor and least common multiple of 5,355 and 40,45? GCD[5355, 40 45] 105 LCM[5355, 40 45] Question 13 Find two ways to compute the sum of the squares of the first 0 consecutive positive integers. Sum[x^, {x, 1, 0}] 870 thirteen = Range[1, 0] {1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15, 16, 17, 18, 19, 0} thirteensq = thirteen^ {1, 4, 9, 16, 5, 36, 49, 64, 81, 100, 11, 144, 169, 196, 5, 56, 89, 34, 361, 400} Total[thirteenSq] 870 Question 14 Find the first three positive solutions to the equation cos(x) = x tan(x). Display your solutions in a graph. f3sols = N[Solve[Cos[x] x * Tan[x] && 0 < x < 7, x, Reals]] {{x }, {x.81704}, {x }} xvalues = { `, `, `}; yvalues = Table[Cos[x], {x, xvalues}]; transpoints = Transpose[{xvalues, yvalues}] {{ , 0.748}, {.81704, }, { , }} pos3points = ListPlot[transPoints, PlotStyle {Red, PointSize[0.03]}]; Page 6 of 7

7 q14plot = Plot[{Cos[x], x * Tan[x]}, {x, 0, 5. Pi}, AxesLabel {x, y}, Ticks {Range[0, Pi, Pi / ], Automatic}, PlotLegends "Expressions" ]; Show[pos3points, q14plot, AxesLabel {x, y}] x tan(x) x cos(x) x tan(x) Question 15 Compute the value of (1/1 + 1/ + 1/3 + 1/4) + (/1 + / + /3 + /4) + (3/1 + 3/ + 3/3 + 3/4) at least two ways. 1./ / a = (1 / / + 1 / / 4) 5 1 b = c = (3 / / + 3 / / 4) 5 4 a + b + c 5 6 * a 5 Page 7 of 7

Arithmetic. Integers: Any positive or negative whole number including zero

Arithmetic. Integers: Any positive or negative whole number including zero Arithmetic Integers: Any positive or negative whole number including zero Rules of integer calculations: Adding Same signs add and keep sign Different signs subtract absolute values and keep the sign of