Chapter 06 (Part II)

1. Chapter 06 (Part II) Functions and an Introduction to Recursion

2. Objective • Write a program for learn C++ subfunction. • Exercise: • Please implement the following functions: • double derivative(double a, double n, double x0); • receives three parameters and returns the slope of axn at x0. • double integral(double a, double n, double x1, double x2); • receives four parameters and returns the area enclosed by axn , y=0, x=x1 and x=x2.

3. Derivative • To compute the slope S between (x1, y1) and (x2, y2): y (x2, y2) y2- y1 (x1, y1) x x2- x1 x1 x2

4. Derivative • To compute the slope of the tangent line at x0: y f(x) x0 x0+dx x0+dx x0+dx x0+dx x

5. Derivative • When dx approaches 0, then the slope of the tangent line at x0 is called the derivative of f(x) at x0. • We would use a very small dx to approximate the derivative.

6. Implementation of derivative() • Program framework prototype implementation

7. The #define Directive • You can use the #define directive to give a meaningful name to a constant in your program. • Example: • DX will be replaced by 0.0001 #include <iostream> #define DX 0.0001 int main () { cout << DX << endl; return 0; } #include <iostream> #define DX 0.0001 int main () { cout << DX << endl; return 0; } 0.0001

8. int S Copy value Output argument Implementation of Subfunction • Remember the only way for a sub-function to communication with outside is through its parameters and return value! int int value 20 x 3 int value = x*x + 2*x + 5; x Parameter Return Value

9. Implementation of derivative()

10. Implementation of derivative() • What the sub-function can only access are its parameters and variables. • Note: do not declare a variable of the same variable name with parameters.

11. Integral • How do we compute the area enclosed by axn , y=0, x=x1 and x=x2? y x x2 x1

12. Integral • Use rectangles of the same width to cover the enclosed field and then sum up their area y f(x1+dx) f(x1) f(x1+2dx) f(x1+5dx) f(x1+3dx) f(x1+4dx) dx dx dx dx dx dx x x2 x1

13. Integral • The measure of area is • The accuracy of area measure depends on how small dx is. y x x2 x1

14. Integral • When dx approaches 0, we can approximate the area below f(x) between x1 and x2. • We denote as the integral of f(x) from x1 to x2.

15. Implementation of integral() • Program framework

16. Implementation of integral() • Use a very small value to represent dx. • Compute the area of rectangle by multiplying f(x) and dx. • Accumulate the area of rectangles. for (double i = x1; i <= x2; i += DX) { …… }