Derivatives, Functions, Integrals, Taylor Series, Limits

Maple session	Comments
> diff(cos(x),x); - sin(x) _______________________________ > diff(cos(x),x,x); - cos(x) _______________________________ > diff(log(sin(x)),x); cos(x) ------ sin(x)	The command `diff` does differentiation. Repeating the variable gives a second derivative. Maple knows a wide variety of standard functions. The names are generally self-evident.
> r := sqrt(x^2+y^2+z^2); 2 2 2 1/2 r := (x + y + z ) _________________________________ > V := proc(x,y,z) 1/r end; V := proc(x,y,z) 1/r end _________________________________ > V := (x,y,z) -> 1/r: _________________________________ > diff(V(x,y,z),x); x - ----------------- 2 2 2 3/2 (x + y + z )	You can define your own functions. The first command assigns the expression $\sqrt{x^2+y^2+z^2}$ to the name r. The second command defines a function V(x,y,z) in terms of r. Notice that it is necessary to specify the dependent variables by including them as arguments to the Maple `proc()` function. The actual statement(s) that define the function are placed between the `proc()` and `end`. The reason for the `end` flag is that function procedures can run on to several lines, and it is necessary to indicate which is the last line. The third command shows the alternative arrow notation that works for simple functions. Notice that the partial derivative is taken in the fourth command.
> diff(V(x,y,z),x,y); x y 3 ----------------- 2 2 2 5/2 (x + y + z )	This is the way to get $\partial^2 V/\partial x \partial y$ .

Maple session	Comments
> int(x^3*cos(x),x); 3 2 x sin(x) + 3 x cos(x) - 6 cos(x) - 6 x sin(x)	Indefinite integral $\int x^3 \cos x dx$
> int(sin(t)/t,t=0..x); Si(x)	Definite integral $\mathop{\rm Si}\nolimits(x) = \int_0^x \sin t dt/t$ . This integral can't be reduced to more standard functions, so it is just called the sine integral and given the name $\mathop{\rm Si}\nolimits(x)$ .
> taylor(",x); 3 5 6 x - 1/18 x + 1/600 x + O(x )	The Taylor series expansion of $\mathop{\rm Si}\nolimits(x)$ at x = 0.
> Order := 10; Order := 10 _______________________________ > taylor("",x); 3 5 x - 1/18 x + 1/600 x - 7 9 10 1/35280 x + 1/3265920 x + O(x )	You can get more terms in the Taylor series by changing the preassigned value of `Order`.
> taylor(sin(x),x=Pi); 3 - (x - Pi) + 1/6 (x - Pi) - 5 6 1/120 (x - Pi) + O((x - Pi) )	Use `x=a` to get the Taylor series expansion about x=a.
> f := proc(x,h) (sin(x+h) - sin(x))/h end; f := proc(x,h) (sin(x+h)-sin(x))/h end __________________________________ > limit(f(x,h),h=0); cos(x)	This is how to find a limiting value. In this case the expression defines the derivative of $\sin x$ at x.

Maple session	Comments
> evalf(erf(1)); .8427007929 ___________________________ > evalf(erf(2)); .9953222650 ___________________________ > limit(erf(x),x=infinity); 1	The error function, known to Maple, is defined as $\begin{displaymath} \mathop{\rm erf}\nolimits(x) = \frac{2}{\sqrt{\pi}}\int_0^x \exp(-t^2) dt \end{displaymath}$