Functions 

We can use expressions in order to work with the kinds of functions we see in calculus. 

>	y:= m*x + b;

 

>	eval(y,x=2);

 

But sometimes the whole expression/eval combination is just too clunky. An alternative is to create a function. 

>	f:= x -> sin(x^2);

 

The above says that f is a rule to map a value to the sine of the square of that value. The rule can be applied to numbers, variables, or expressions, with no need to use eval. 

>

f(1);

 

>

f(t);

 

>	f(sqrt(t));

 

If you have an expression and want to make it a function, use: 

>	g:= unapply( p, x );

 

Conversely, g(x) is equivalent to the expression p defined earlier. You can easily go back and forth and use whichever is most convenient. 

Limits 

You can take limits of expressions. 

>	limit( x^2-1, x=2 );

 

>	limit( 1/x, x=infinity );

 

>	limit( x*sin(1/x), x=0 );

 

>	limit( (sin(t+h)-sin(t))/h, h=0 );

 

>

 

Differentiation 

Taking one or more derivatives of a function is easy.  

>	diff( sin(t), t );

 

>	diff( sin(t), t,t );

 

If you have a function, you can convert it to an expression first. 

>

f(t);

 

>	diff( f(t), t );

 

Alternatively, you can use D. The result is another function representing the derivative. This makes it easy to find values at different arguments.  

>	fprime:= D(f);

 

Here are two ways to find a derivative at a point. 

>	eval( diff(f(x),x), x=1 );

 

>

D(f)(1);

 

The derivative of an undefined symbol is a purely symbolic result. Observe the difference in these next two statements: 

>	diff( g(t), t );

 

>	diff( g, t );

 

You must make the dependence on the independent variable explicitly clear.   

Integration 

You can do indefinite or definite integration. In the indefinite case, Maple does not add the integration constant! 

>	int( sin(theta), theta );

 

>	int( sin(theta), theta=0..Pi/2 );

 

If you capitalize Int, Maple just spits back the integral without trying to evaluate it. Then, you can use value to ask it to find the integral. This is a useful habit, to make sure you are getting the integral you think you asked for.  

>	Int( sin(theta), t=0..Pi/2 );   # oops!

 

>

value(%);

 

>	Int( sin(theta), theta=0..Pi/2 );

 

>

value(%);

 

Sometimes Maple is actually wrong. 

>	int( sin(n*theta), theta );

 

What are we to make of that last result if n=0? You could answer that the original problem is very easy in that case. But here is another one where it is not. 

>	int( x^(n-1), x );

 

As you know, most integrals are hard. Many are impossible. If Maple can't get the answer, it just returns the unevaluated form. 

>	int( sin(cos(theta)), theta );