Housekeeping Commands

>

restart;

>	with(plots, display, implicitplot):

>	with(plottools, disk):

Section 4.1 - Maple and limit and Limit

Here are some limits which don't present as difference quotients:

>	CompInt := n -> (1+1/n)^n;

>	FarOut := Limit( CompInt(k), k=infinity);   "Inert form"

>	value(FarOut);    "value" helps evaluate "algebraically"

>	evalf(%, 14);

################################################################

>	divot := x -> (x^3-125)/(x-5);

>

divot(5);

Error, (in divot) numeric exception: division by zero

>	limit( divot(x), x=5);   Flying a bit blinded.

################################################################

>	InDet := n -> (1-3/n)^n;

>	Limit( InDet(i), i = infinity);

>

value(%);

#################################################################

Here's a difference-quotient limit such as appears on page 55-56:

>	NQ := (f,t,k) -> (  f(t+k) - f(t)  )/k;

>	NQ(sin,Pi/2,Pi);  NQ(exp,-1/100,2/100);  evalf(%);

>	gg := x -> (x+3)/(x-5);

>	Limit( NQ(gg,x,h), h=0);

>	DerivExpr := value(%);

>	ggPRIME := unapply(DerivExpr, x);

>	POTx := 2:

>

>	POT := [POTx, gg(POTx)];

>	M_tan := ggPRIME( POT[1] );

>	TanLinEqn := y = POT[2] + M_tan*( x - POT[1] );

#################################################################

Now build the sequence of plot structures to show tangency:

>	P := disk( POT, 1/25, colour=BLACK):

>	delta := 3/2;  epsilon := 1;

>	XRG := POT[1]-delta..POT[1]+delta;

>	YRG := POT[2]-epsilon..POT[2]+epsilon;

>	P := P, plot(gg, XRG, YRG):

>	P := P, implicitplot(TanLinEqn, x=XRG, y=YRG, colour=black):

>	display([P], scaling=CONSTRAINED);

Section 4.2 on Derivatives of Maple Functions

>

gg(x);

>	ggp := D(gg);  ggpp := D(ggp);

>	simplify( ggp(t) );

>	TangentLineSlope := ggp(POTx);

##################################################################

>

D(p*q);

>	D(p/q);  simplify(%);

>

D(p^2);

>

D(p@q);

>	(D@@2)(p*g);  (D@@3)(p*g);

#####################################################################

>	LISTfun := [ln,sin,cos,x -> gg(2*x),exp, t -> exp(2*t)*cos(3*t)];

>	LISTders := map(D, LISTfun);

>	LISTders[nops(LISTders)](0);

>

4.3 on derivatives of Maple expressions

>	fexpr := x*sin( exp(3*x) );

>	diff(fexpr,x);

4.4 - Maple Implicit Differentiation is Different Nowadays

As with LeastSquares, Maple has done some work to ease implicit differentiation, so section 4.4 of our text is a bit out of date.  Here we demo the new "implicitdiff" command:

>	CUBIC := x^3 + y^3 = 6*x*y - 3;   Slightly different from page 61.

>	yPRIME := implicitdiff(CUBIC,y,x);

>	yp := unapply(yPRIME,x,y);

>	TanSlope := yp(1,2);

>	TanLineEqn := y-2 = TanSlope*(x-1);

>	implicitplot([CUBIC,TanLineEqn],x=-3..3,y=-3..3,colour=[BLACK,RED]);

Assignment #6

Here are the problems for Assignment #6, due Thursday, 10/13/05.

Page 66:

1(c) --  Be sure to use  as they say.

3  -- make a blob for the point of tangency as well.

5

6

12

16 -- use "implicitdiff" rather than the text's methods

>	##############################################################