Some Functions

• In programming a function is a small section of code
  • It possibly takes some input known as parameters/arguments.
  • It possibly returns a value.
  • If you have the definition of a function in mathematics, ignore it.
• We will discuss creating functions later.
• In this section we will discover a number of mathematical functions.
  • Most are from the math library
  • #include <cmath>
• The power function
  • float pow(float base, float exponent)
    • This is called the function prototype
    • It tells us how to use the function.
  • computes base^exponent where
  • We call base and exponent parameters
  • Consider the following code to compute continuously compounded interest: $P(t) = P_0e^{rt}$

    • //   3000 invested at 7% compounded continuously for 5 years
      float amount{3000};
      float interestRate{0.07}
      float time{5}
      float finalAmount{0}; 
      float exponent{0};
      
      exponent = interestRate * time;
      
      finalAmount = amount * pow(M_E, exponent); 

    • M_E comes from cmath and is $e$
    • M_E and exponent are called arguments.
  • See this reference
    • Note there are at least four different versions.
    • the compiler selects the version to call based on the arguments.
• abs and fabs and labs
  • abs reference
  • Compute the absolute value of an integer, (float, double, long double) and long
• sin, cos ...
  • Trig functions.
  • Works with angles in radians, NOT DEGREES.
  • But M_PI is defined in cmath.
• sqrt
  • Square root.
• Many others
• Problem 3 from the end of the chapter

  The factorial of a number n, (written n!) is the number times the factorial of itself minus one. This self-referential definition is easier to understand through an example: The factorial of 2 is 2 * 1; the factorial of 3 is 3 * 2 * 1; the factorial of 4 is 4 * 3 * 2 * 1; and so on. Factorials grow very large, very quickly. An approximation to the factorial for larger values is given by Stirling's formula: $$n! = e^{-n}n^n\sqrt{2\pi n}$$ The exp function in the <cmath> header file gives the value of $e$ raised to a given power. Write a c++ program that computes the factorial of 15 both directly and with Stirling's formula, and outputs both results, together with their difference. You will need to use the double type for this computation.