var NP = 20; var probdat = new Object; var adat = new Object; var bdat = new Object; var cdat = new Object; var ddat = new Object; var hints = new Object; var tested = new Array(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0); var pm = new Array(1,2,3,4); var mp = new Array(1,2,3,4); var ans = new Array(11,10,10,7,12,10,5,5,3,10,4,5,4,6,0,6,13,12,1,9); var hn = new Array(0,0,1,2,3,0,4,5,6,2,-7,8,9,10,11,12,13,14,15,16,17,18,19,2,20,21,22,23,24,1,25,26,27,2,28,29,-30,31,32,0,33,34,35,36,2,37,38,13,14,19,39,14,40,41,31,13,-42,-43,13,13,44,45,46,47,39,48,49,41,18,18,50,51,52,-53,54,55,2,56,57,39); hints[0] = "See The Extreme Value Theorem ( p. 275 )"; hints[1] = "Try f(x)=-x"; hints[2] = "See The Intermediate Value Theorem ( p. 127 )"; hints[3] = "Try

"; hints[4] = "Try f(x)=x+4"; hints[5] = "The product of two continuous functions is a continuous function ( see Theorem 4, p.123 )"; hints[6] = "Try

"; hints[7] = "Take

Then g(5.5) is undefined"; hints[8] = "The sum of two continuous functions is a continuous function ( see Theorem 4, p. 123 )"; hints[9] = "It's obvious"; hints[10] = "The sum of two increasing functions is an increasing function"; hints[11] = "The quotient of two continuous functions is a continuous function on its domain ( see Theorem 4, p.123;

on [4,10] )"; hints[12] = "Take f(x)=x-5"; hints[13] = "Take f(x)=x"; hints[14] = "Try f(x)=x²"; hints[15] = "(0,10] in not closed! Try

"; hints[16] = "(0,10] is not closed! Try

"; hints[17] = "Take

"; hints[18] = "Example: f(x)=1 for all x"; hints[19] = "Take f(x)=e^x"; hints[20] = "Since

we have

and f(0)=0. See the Squeeze Theorem ( p. 117 )"; hints[21] = "Take

. Then

- not continuous at x=0"; hints[22] = "

"; hints[23] = "f(0)=0 is a minimum"; hints[24] = "-f is increasing and g is a sum of two increasing functions" hints[25] = "The quotient of two continuous functions is a continuous function on its domain ( see Theorem 4, p.123 )"; hints[26] = "Take f(x)=2-x"; hints[27] = "Take g(x)=f(x)-x and apply the Intermediate Value Theorem ( p. 127 ) for g"; hints[28] = "Take f(x)=x²-6x+7"; hints[29] = "Take

"; hints[30] = "

and

are undefined"; hints[31] = "See Definition 2 p. 122"; hints[32] = "Take

"; hints[33] = "Take

"; hints[34] = "Take g(x)=1 for all

then f(x)=x"; hints[35] = "f(0)=0"; hints[36] = "Take g(x)=1 for all

then f(x)=x"; hints[37] = "Take

"; hints[38] = "

so by the Squeeze Theorem ( p. 117 ),

"; hints[39] = "See Theorem 9, p. 126"; hints[40] = "Take f(x)=x+1"; hints[41] = "See Theorem 4, p.123"; hints[42] = "

- closed interval"; hints[43] = "

"; hints[44] = "Let

"; hints[45] = "g is a constant"; hints[46] = "

"; hints[47] = "Take

"; hints[48] = "Take f(x)=x-1"; hints[49] = "g(1)=0"; hints[50] = "Otherwise we would have f(-1)<f(1)"; hints[51] = "

and f are continuous at

"; hints[52] = "Use the Squeeze Theorem"; hints[53] = "h(x) is undefined for any x"; hints[54] = "Take f(x)=1-e^-x²" hints[55] = "Take

"; hints[56] = "Take f(x)=2x-6"; hints[57] = "Try f(x)=-2x+6"; probdat[1] = "f is continuous on [2,7]"; adat[1] = "f has a maximum on [2,7]" bdat[1] = "f has a minimum on [2,7]" cdat[1] = "f(2)<f(7)" ddat[1] = "if c is between f(2) and f(7), then there exists d in [2,7] such that f(d)=c"; probdat[2] = "f is continuous on [-3,7]" adat[2] = "

" bdat[2] = "f has a minimum on [-3,7]" cdat[2] = "f has a root on [-3,7]" ddat[2] = "

" probdat[3] = "f is continuous on [2,9], f(2)=3, f(9)=-7"; adat[3] = "f is decreasing on [2,9]"; bdat[3] = "f has a root in [2,9]"; cdat[3] = "the function

is continuous on [2,9]"; ddat[3] = "

"; probdat[4] = "f is continuous and increasing on [4,10]"; adat[4] = "f(4) is the minimum of f"; bdat[4] = "g(x)=x+f(x) is increasing on [4,10]"; cdat[4] = "

is continuous on [4,10]"; ddat[4] = "f has no root on [4,10]"; probdat[5] = "f is continuous on (0,10]"; adat[5] = "

does not exist"; bdat[5] = "

is not continuous at x=0"; cdat[5] = "f need not have a maximum"; ddat[5] = "f need not have a minimum"; probdat[6] = "f is continuous on

"; adat[6] = "

"; bdat[6] = "f need not have a root on

" cdat[6] = "f must have a minimum"; ddat[6] = "if f(2)=2 and f(5)=8 then there exists d such that f(d)=4"; probdat[7] = "f is continuous and

for every x"; adat[7] = "

, g(0)=0, is continuous at x=0"; bdat[7] = "

, h(0)=0, is continuous at x=0"; cdat[7] = "f(x) has a root"; ddat[7] = "f need not have a minimum"; probdat[8] = "f is continuous and decreasing on [-1,3]"; adat[8] = "g(x)=x-f(x) is increasing on [-1,3]"; bdat[8] = "

for all

"; cdat[8] = "

"; ddat[8] = "f has a root on [-1,3]"; probdat[9] = "f is continuous on [1,3] and f(1)=2, f(3)=-2"; adat[9] = "The equation x=f(x) has at least one root"; bdat[9] = "g(x)=xf(x) has a root on [1,3]"; cdat[9] = "

"; ddat[9] = "

for all

" probdat[10] = "f is continuous on

"; adat[10] = "

is continuous on

"; bdat[10] = "

"; cdat[10] = "

"; ddat[10] = "f has a maximum on

"; probdat[11] = "g is defined on [-1,1], f(x)=xg(x) is continuous on [-1,1]"; adat[11] = "g is continuous on [-1,1]"; bdat[11] = "g has a root on [-1,1]"; cdat[11] = "f has a root on [-1,1]"; ddat[11] = "f has a maximum on [-1,0)"; probdat[12] = "f is continuous on [-2,8)"; adat[12] = "There exists

such that

"; bdat[12] = "f has a maximum on [-2,8)"; cdat[12] = "

exists"; ddat[12] = "f is not increasing"; probdat[13] = "f is continuous on

"; adat[13] = "f does not have a minimum"; bdat[13] = "f has a root"; cdat[13] = "

is continuous"; ddat[13] = "f(-1)<0<f(1)"; probdat[14] = "f is continuous at x=0"; adat[14] = "

"; bdat[14] = "g(x)=x²f(x) is continuous at x=0"; cdat[14] = "

"; ddat[14] = "f is not continuous at x=1"; probdat[15] = "f is continuous on

"; adat[15] = "f need not have a maximum on

"; bdat[15] = "f is not continuous at x=0"; cdat[15] = "f does not have a root on

"; ddat[15] = "

"; probdat[16] = "f is continuous on

and f(0) is defined"; adat[16] = "f is continuous on [-1,1]"; bdat[16] = "g(x)=f(0) is continuous on [-1,1]"; cdat[16] = "f is continuous on (-1,0)"; ddat[16] = "There exists d, -1<d<1, such that f(d)=1"; probdat[17] = "f is continuous on [1,2]"; adat[17] = "e^f(x) is continuous on [1,2]"; bdat[17] = "

is continuous on [1,2]"; cdat[17] = "g(x)=xf(x)-f(x) has a root on [1,2]"; ddat[17] = "

has a maximum on [1,2]"; probdat[18] = "f is continuous on [-1,1], f(-1)=f(1)"; adat[18] = "There exists

such that

"; bdat[18] = "f has a root on [-1,1]"; cdat[18] = "f cannot be strictly increasing"; ddat[18] = "

"; probdat[19] = "

for all

"; adat[19] = "g(x)=xf(x) is continuous at x=0"; bdat[19] = "

is continuous"; cdat[19] = "f has a maximum"; ddat[19] = "

exists"; probdat[20] = "f is continuous on [2,4], f(2)=-f(4)"; adat[20] = "f has a root on [2,4]"; bdat[20] = "f cannot be increasing"; cdat[20] = "f(2) is negative"; ddat[20] = "f(2x) in continuous on [1,2]"; var existWin2 = 0; var num = 1; var cp = 0; var good = 0; function DisplayProblem(problem) { if ( existWin2==0 ) { win2 = window.open("","Problem","location=no,directories=no,status=yes,menubar=no,scrollbars=yes,resizable=yes,copyhistory=yes,width=550,height=500"); } win2.focus(); if ( problem==0 ) { cp++; with(Math) { num=floor(NP*random())+1; for ( i=0; i<5; i++ ) { m1 = floor(4*random()); m2 = floor(4*random()); temp = pm[m1]; pm[m1] = pm[m2]; pm[m2] = temp; } for ( i=0; i<4; i++ ) { mp[pm[i]-1]=i+1; } } if ( tested[num-1] ) { for ( i=0; iNP ) { num=1; } if ( !tested[num-1] ) { break; } } if ( i==NP ) { DisplayFinalPage(); return; } } } else { num = problem; } // tested[num-1] = 1; win2.document.open(); win2.document.writeln(""); win2.document.write("

Problem "+cp+". "); win2.document.write(probdat[num]); win2.document.writeln("

Please select all statements that are always true.

"); win2.document.writeln("

"); win2.document.write("

"); win2.document.write(" A. "); DisplayChoice(pm[0],num,0); win2.document.write(" B. "); DisplayChoice(pm[1],num,0); win2.document.write(" C. "); DisplayChoice(pm[2],num,0); win2.document.write(" D. "); DisplayChoice(pm[3],num,0); win2.document.writeln("

"); win2.document.writeln("

"); win2.document.close(); existsWin2 = 1; } function DisplayChoice(i,num,chk) { if ( i==1 ) { win2.document.write(adat[num]); } else if ( i==2 ) { win2.document.write(bdat[num]); } else if ( i==3 ) { win2.document.write(cdat[num]); } else { win2.document.write(ddat[num]); } if ( chk==1 ) { win2.document.write("

"); good++; } else if ( chk==2 ) { win2.document.write("

"); } win2.document.writeln("

"); } function bit(x,k) { return ( (x>>k) & 1 ); } function DisplayHint(i1,n) { var hind = hn[(n-1)*4+i1-1]; win2.document.writeln(""); if ( bit(ans[n-1],i1-1)==1 ) { win2.document.writeln("THIS IS TRUE! "); } else if ( hind>=0 ) { win2.document.writeln("THIS IS NOT ALWAYS TRUE! "); } else { win2.document.writeln("THIS IS NEVER TRUE! "); hind = -hind; } win2.document.writeln(""); win2.document.write(hints[hind]); win2.document.writeln(".

"); } function DisplaySolution(form) { if ( tested[num-1]!=0 ) { DisplayProblem(0); return; } tested[num-1] = 1; var num1=0; for ( i=3; i>=0; i-- ) { num1 = num1*2; if ( form.test[mp[i]-1].checked ) { num1 = num1+1; } } win2.document.open(); win2.document.writeln(""); if ( num1==ans[num-1] ) { win2.document.writeln("

YOU ARE RIGHT!!!

"); win2.document.write("

Problem "+cp+". "); win2.document.write(probdat[num]); win2.document.writeln("

Your answer:

"); win2.document.writeln("

"); win2.document.write("

"); win2.document.write("

A. "); DisplayChoice(pm[0],num,1); win2.document.write("

B. "); DisplayChoice(pm[1],num,1); win2.document.write("

C. "); DisplayChoice(pm[2],num,1); win2.document.write("

D. "); DisplayChoice(pm[3],num,1); win2.document.writeln("

"); win2.document.write("

:"); win2.document.write(""); win2.document.write("Click here to proceed."); win2.document.writeln("

"); } else { win2.document.writeln("

HERE IS WHAT YOU'VE GOT WRONG:

"); win2.document.write("

Problem "+cp+". "); win2.document.write(probdat[num]); win2.document.writeln("

Your answer:

"); win2.document.writeln("

"); win2.document.write("

"); win2.document.write("

A. "); if ( bit(num1,pm[0]-1)==bit(ans[num-1],pm[0]-1) ) { DisplayChoice(pm[0],num,1); } else { DisplayChoice(pm[0],num,2); DisplayHint(pm[0],num); } win2.document.write("

B. "); if ( bit(num1,pm[1]-1)==bit(ans[num-1],pm[1]-1) ) { DisplayChoice(pm[1],num,1); } else { DisplayChoice(pm[1],num,2); DisplayHint(pm[1],num); } win2.document.write("

C. "); if ( bit(num1,pm[2]-1)==bit(ans[num-1],pm[2]-1) ) { DisplayChoice(pm[2],num,1); } else { DisplayChoice(pm[2],num,2); DisplayHint(pm[2],num); } win2.document.write("

D. "); if ( bit(num1,pm[3]-1)==bit(ans[num-1],pm[3]-1) ) { DisplayChoice(pm[3],num,1); } else { DisplayChoice(pm[3],num,2); DisplayHint(pm[3],num); } win2.document.writeln("

"); win2.document.write("

:"); win2.document.write(""); win2.document.write("Click here to proceed."); win2.document.writeln("

"); } win2.document.close(); } function closeWin() { onerror=null if ( existsWin2==1 ) {win2.close()} } function percentage(x,y) { return Math.round(1000.0*(x/y))/10.0; } function DisplayFinalPage() { for ( i=0; i"); win2.document.write(" You have reached the end of this quiz."); win2.document.writeln(" Thank you!

"); win2.document.writeln("You answered correctly "+good+" questions out of "); win2.document.writeln((NP*4)+" possible which makes about "+percentage(good,NP*4)+"%

"); win2.document.write(""); win2.document.write("Click here to run the test again.

"); win2.document.write(""); win2.document.write("Click here to quit."); win2.document.close(); }