Auxiliary Procedures -- do not display

>	restart; with( plots ): with( Student[Calculus1] ):

Warning, the name changecoords has been redefined

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figure := proc( theta )   local A, C, P, S, T, Title;   C := implicitplot( x^2+y^2=1, x=-1..1, y=-1..1, tickmarks=[0,0] ):   S := theta -> plot( [[0,0],[cos(theta),sin(theta)],[cos(theta),0]], color=blue ):   A := theta -> plot( [cos(theta)*cos(t),cos(theta)*sin(t),t=0..theta], color=green ):   P := theta -> pointplot( [ [1,0],                            [cos(theta),0],                            [cos(theta)*cos(theta),cos(theta)*sin(theta)],                            [cos(theta),sin(theta)] ],                            symbol=circle, symbolsize=12 ):   T := theta -> textplot( {                   [-0.1,-0.1,"O"],                   [1.25,-0.1,"A (1,0)"],                   [cos(theta),-0.1,"B"],                   [cos(theta)*cos(theta)-0.1,cos(theta)*sin(theta)+0.1,"C"],                   [cos(theta)+0.25,sin(theta)+0.1,"P (cos t, sin t)"],                   [-0.2,1.1,"(0,1)"],                   [0.3,0.1,"t"] } ):     Title := proc( theta )       if nargs > 1         then           if args[2]=`shorttitle`             then return sprintf( "Unit circle with theta=%07.5f", theta )             else return ""           end if         else           return sprintf( "  When t=%07.5f:\narea( sector(COB) ) =   %07.5f\narea( triangle(POB) ) = %07.5f\narea( sector(POA) ) =   %07.5f",                             theta,                             abs(theta)*cos(theta)^2/2,                             cos(theta)*abs(sin(theta))/2,                             abs(theta)/2 ):       end if     end proc;   plots[display]( [C,S(theta),A(theta),P(theta),T(theta)],                   view=[-1.5..1.5,-1.5..1.5],                   title=Title(args),                   scaling=constrained ): end proc:

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# Epsilon-Delta Maplet # Copyright 2002 Waterloo Maple Inc. # This maplet graphically illustrates the delta-epsilon definition of the limit.  The definition states that limit( f(x), x=a ) = L if, for all e>0, there exists d>0 such that 0<|x-a|<d implies |f(x)-L|<e.   # # Enter a function f(x) and values for a, L, e and d.  You can then experiment with the values of a, e and d to try and find values that satisfy the definition by seeing the plot of f(x) around x=a annotated with these values.  Note that d satisfies the definition as long as the blue region fits between the two red horizontal lines on the plot. # # In the plot, x=a is marked by a gray vertical line, and the value L by a gray horizonal line. # # The interval [a-d, a+d] is marked by a blue line on the x-axis. # # The range of function values implied by the choice of d is marked by a blue region. # # The two red horizontal lines indicate y=L+e and y=fL-e. # # To run this maplet, click the !!! button in the tool bar. # #restart: with(LinearAlgebra): with(plots): with(plottools): with(Maplets[Elements]): ############################################################ # the drawing procedure cgraph :=  proc(f, a, L, ep, deltav, xmin, xmax, ymin, ymax)   local fc, g, aline, Lline, e, ept1, ept2, elinem, elineu, elinel, asd,         aad, dptr1, dptr2, dliner, dptb1, dptb2, dlineb, j, k, p1, p2, delta;   #plotting the graph   fc := plot(f, x=xmin..xmax, y=ymin..ymax, color=navy);   #drawing a vertical and horizonal line though (a,f(a))   g := unapply(f, x):   aline := line([a, 0], [a, L], thickness=2, color=gray):   Lline := line([xmin, L], [xmax, L], color=gray):   #scaling the value from the epsilon-slider to between 0 and 1   e := ep/100:   #drawing the epsilon neighborhood and horizonal lines at the epsilon end-points   ept1 := [0, L-e]:   ept2 := [0, L+e]:   elinem := line(ept1, ept2, thickness=4, color=red):   elineu := line([xmin, L+e], [xmax, L+e], color=red):   elinel := line([xmin, L-e], [xmax, L-e], color=red):   #scaling the value from the delta-slider to between 0 and 1   delta := deltav/100:   asd := a-delta:   aad := a+delta:   #the shaded in parts   p1 := polygon([[asd, 0], [asd, g(aad)], [aad, g(aad)], [aad, 0]],                  color=turquoise, style=patchnogrid):   p2 := polygon([[0, g(aad)], [0, g(asd)], [aad, g(asd)], [aad, g(aad)]],                  color=turquoise, style=patchnogrid):   #drawing the vertical lines from a+-delta to the function   dptr1 := [asd, 0]:   dptr2 := [aad, 0]:   dliner := line(dptr1, dptr2, thickness=4, color=blue):   dptb1 := [0, g(asd)]:   dptb2 := [0, g(aad)]:   plots[display](aline, Lline, fc, elinem, elineu, elinel, dliner, p1, p2): end proc: ############################################################ # converts the value chosen on the slider into the real value # chosen for epsilon and delta realvalue := proc(rvalue)   evalf[3](rvalue/100.0): end proc:

Warning, the name arrow has been redefined

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f := 3*sqrt(x)/(x+1): a := 3: L := limit( f, x=a ): plotED := cgraph(f, a, L, 12, 50, 0, 4, 0, 2): plotL1 := textplot( {[a-0.25,-0.1,"d"], [a+0.25,-0.1,"d"] }, font=[SYMBOL,18], color=blue ): plotL2 := textplot( {[-0.2,L-0.12,"e"], [-0.2,L+0.12,"e"]}, font=[SYMBOL,18], color=red ): plotL3 := textplot( {[a,-0.1,"a"], [-0.2,L,"L"]}, font=[TIMES,ITALIC,18] ): R := display( [plotED, plotL1, plotL2, plotL3], view=[-0.2..5,-0.2..1.7], tickmarks=[4,3], labels=["",""], title=sprintf("An epsilon-delta Graph for\nlimit( %a, x=%a ) = %a", f, a, L) ):

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