1. Computer Science 1620 Math Library

2. Remember this program? suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years

3. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years #include using namespace std; int main() { cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl; return 0; }

4. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years #include using namespace std; int main() { double balance = 25000.0; balance = balance * 1.08; cout << "Year 1: $" << balance << endl; balance = balance * 1.08; cout << "Year 2: $" << balance << endl; balance = balance * 1.08; cout << "Year 3: $" << balance << endl; return 0; }

6. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl; return 0; }

7. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) one year, b) two years c) three years #include using namespace std; int main() { double balance = 25000.0; cout << fixed << showpoint << setprecision(2); balance = balance * 1.08; cout << "Year 1: $" << balance << endl; balance = balance * 1.08; cout << "Year 2: $" << balance << endl; balance = balance * 1.08; cout << "Year 3: $" << balance << endl; return 0; }

9. Suppose we alter the program? suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

10. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl; return 0; }

11. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 1: $" << 25000 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl; return 0; }

12. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 2: $" << 25000 * 1.08 * 1.08 << endl; cout << "Year 3: $" << 25000 * 1.08 * 1.08 * 1.08 << endl; return 0; }

13. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 20: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 30: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; return 0; }

14. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { double balance = 25000.0; cout << fixed << showpoint << setprecision(2); balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 10: $" << balance << endl; balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 20: $" << balance << endl; balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 30: $" << balance << endl; return 0; }

16. The previous example worked, but … to calculate 1.08 30 required that we manually type in thirty multiplications suppose we had wanted to know the balance after 100 years?

17. pow(b,e) to compute b raised to the power of e, one can use the pow function syntax: the pow statement itself is an expression value: the first expression raised to the power of the second expression pow(, ) numeric expression numeric expression

18. Example: #include using namespace std; int main() { cout << pow(3.0, 3.0) << endl; cout << pow(3, 4.0) << endl; cout << pow(4.0, 3) << endl; int x = 5; double y = 6.0; cout << pow(x, y) << endl; cout << pow(x, y) + pow(3.0, 3.0) << endl; cout << pow( pow(x, y), pow(3.0, 3.0) ) << endl; return 0; }

19. A note about pow you cannot send it two integer expressions cout << pow(3, 4) << endl; // compiler error this has to do with function overloading we will examine functions later for now, just know that one expression has to be a floating point type

20. The cmath library FunctionDescription abs(x)Returns the absolute value of x pow(x, y)Returns x y sqrt(x)Returns the square root of x sin(x)Returns the sine of x cos(x)Returns the cosine of x tan(x)Returns the tangent of x log(x)Returns the natural logarithm of x ln x log10(x)Returns log 10 x exp(x)Returns e x floor(x)Returns the closest whole number < x ceil(x)Returns the closest whole number > x

21. A Note about Types when using a math function (from previous slide), the type of the expression is the same as the type of the input expression, with some exceptions: an integer expression is converted to a double for the pow function, the type of the expression is the largest of the base and exponent expressions for abs integers are not converted to doubles shorts and chars are converted to ints

22. ExpressionType sqrt(2.0)double sin(2.0f)float cos(2)double log(3 + 4.0f)float exp(2) + exp(2.0)double abs(75);int floor(3.5);double

23. A Note about the Trig Functions the input expression is considered to be radians hence, sin(30.0) = -0.988032, not 0.5 to convert radians to degrees, multiply the value by (π / 180)

24. Example: #include using namespace std; int main() { const double PI = 3.14159; cout << sin(30) << endl; cout << sin(30 * PI / 180) << endl; return 0; }

26. Back to our previous example suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years

27. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 20: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; cout << "Year 30: $" << 25000 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 << endl; return 0; }

28. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { cout << fixed << showpoint << setprecision(2); cout << "Year 10: $" << 25000 * pow(1.08, 10) << endl; cout << "Year 10: $" << 25000 * pow(1.08, 20) << endl; cout << "Year 10: $" << 25000 * pow(1.08, 30) << endl; return 0; }

29. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { double balance = 25000.0; cout << fixed << showpoint << setprecision(2); balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 10: $" << balance << endl; balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 20: $" << balance << endl; balance = balance * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08 * 1.08; cout << "Year 30: $" << balance << endl; return 0; }

30. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { double balance = 25000.0; cout << fixed << showpoint << setprecision(2); balance = balance * pow(1.08, 10); cout << "Year 10: $" << balance << endl; balance = balance * pow(1.08, 10); cout << "Year 20: $" << balance << endl; balance = balance * pow(1.08, 10); cout << "Year 30: $" << balance << endl; return 0; }

31. Suppose that I invest $25000 into a mutual fund that returns 8% per year. Write a program to calculate how much the fund will be worth after a) ten years, b) twenty years c) thirty years #include using namespace std; int main() { double balance = 25000.0; double increase = pow(1.08, 10); cout << fixed << showpoint << setprecision(2); balance = balance * increase; cout << "Year 10: $" << balance << endl; balance = balance * increase; cout << "Year 20: $" << balance << endl; balance = balance * increase; cout << "Year 30: $" << balance << endl; return 0; }

32. Example 2: Suppose you have a cannon. Given an initial velocity of a cannonball and the angle of the cannon θ, output (1) the time that the ball remains in the air and (2) the maximum height of its trajectory

33. Arithmetic Data and Operators Given a value θ and initial velocity v 0, we can calculate the time that the ball stays in the air: and its maximum height: g is approximately -9.81 m/s at sea level

34. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

35. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

36. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

37. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

38. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

39. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

40. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / -9.81; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; } cmath call

41. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / -9.81; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; } Magic Number

42. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; } Expects radians

43. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * 3.14159 / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; } Magic Number

44. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

45. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

46. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

47. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

48. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(2); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }

49. Write a program that accepts an initial velocity and the angle of trajectory, and computes the hangtime and the maximum height of the cannonball. Output to three decimal places. #include using namespace std; int main() { const double PI = 3.14159; const double G = -9.81; double v0; cout << "Initial velocity (m/s): "; cin >> v0; double theta; cout << "Angle of trajectory (degrees): "; cin >> theta; theta = theta * PI / 180.0; double time = -2 * v0 * sin(theta) / G; double height = -pow(v0 * sin(theta), 2) / (2 * G); cout << fixed << showpoint << setprecision(3); cout << "Hangtime (s): " << time << endl; cout << "Height (m): " << height << endl; return 0; }