1. Chapter 11 Calculus

2. Symbolic Expressions Required: > Symbolic Math Toolbox > Use Symbolic Variables

3. Functions for Symbolic Processing x = sym(‘x’) Creates the symbolic variable with name x. syms x y u v Creates the symbolic variables x, y, u, & v. Simplify(ans) Simplifies the expression ans.

4. Symbolic Expression Example >>syms x y >>s=x+y; >>r=sqrt(x^2+y^2);

5. Symbolic Expression Example >>n=3; >>syms x; >>A=x.^((0:n)’*(0:n)) >>A= [1, 1, 1, 1] [1, x, x^2, x^3] [1, x^2, x^4, x^6] [1, x^3, x^6, x^9]

6. Manipulating Expressions Use the expand command: >>syms x y >>expand((x+y)^2) ans= x^2+2*x*y+y^2 >>expand(sin(x+y)) ans= sin(x)*cos(y)+cos(x)*sin(y)

7. Evaluating Expressions Use subs(E, old, new) or double(y): >>syms x >>E=x^2+6*x+7; >>G=subs(E,x,2) G= 23

8. Multiple Variables >>syms x y z >>E=x^2+6*y+2*z; >>subs(E,{x,y,z},{2,2,3}) ans = 22

9. Using Double Command Example >>sqroot2=sym(‘sqrt(2)’); >>y=6*sqroot2 y= 6*2^(1/2) >>z=double(y) z= 8.4853

10. Plotting Expressions Use ezplot command: >>syms x >>E=x^2-6*x+7; >>ezplot(E,[-2 6])

12. Symbolic Calculus Functions diff(E) Returns the derivative of the expression E with respect to the default independent variable. diff(E,v) To variable v. diff(E,v,n) n th derivative int(E) Returns the integral limit(E) Returns the limit

15. Laplace Transforms laplace(exp) Returns Laplace transform. ilaplace(exp) Returns inverse Laplace.