All of the Maple commands necessary for this lab have already been seen before.

- Find where
*y*=*cos*(*x*) and intersect.> solve(cos(x)=x^2+2*x-3,x);

> fsolve(cos(x)=x^2+2*x-3,x);

> plot({cos(x),x^2+2*x-3},x=-5..5);

> fsolve(cos(x)=x^2+2*x-3,x=-5..0);

These two functions intersect at approximately

*x*=1.1085 and*x*=-2.7534. This can be viewed on the graph. - Find the first order partial derivatives for the following
function.
Recall that there are multiple ways of doing this. These are two of the ways seen in the past.

- The first way is using the
`diff`command.> f:=(x,y)->x^2*y^2+y*sin(x);

> f_x:=diff(f(x,y),x);

> f_y:=diff(f(x,y),y);

- The second method is using the
`D`command. Notice that the number in the square braces corresponds to the variable in that position when defining the function.> f_x:=D[1](f);

> f_y:=D[2](f);

- The first way is using the
- Compute this integral
over the region

*R*where and> inner:=int(2*x*y/(x^2+1),x=0..1);

> outer:=int(inner,y=1..3);

> evalf(outer);

The exact value of this integral is . The decimal approximation to this integral is 2.7726. Note that this integral could be computed in one step.

> int(int(2*x*y/(x^2+1),x=0..1),y=1..3);

- Compute the following double integral.
> inner:=int(exp(y-x),y=x..2*x);

> outer:=int(inner,x=0..1);

> evalf(outer);

The exact value of this integral is

*e*-2. The decimal approximation to this integral is .7183. Or to save some steps, the following Maple command can be executed.> int(int(exp(y-x),y=x..2*x),x=0..1);

Mon Apr 21 09:58:06 EDT 1997