var NP = 15; 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); var pm = new Array(1,2,3,4); var mp = new Array(1,2,3,4); var ans = new Array(5,11,11,2,13,9,3,5,4,7,1,3,6,5,10); var hn = new Array(1,-2,3,4,5,3,6,7,8,9,10,11,12,13,12,14,15,16,17,18,19,20,21,22,23,24,24,23,25,26,27,28,29,30,31,32,33,34,35,33,36,37,38,39,25,40,41,42,43,44,45,44,46,47,48,49,-50,51,52,53); hints[1] = "

is continuous, so

"; hints[2] = "f is decreasing on (2,3)"; hints[3] = "f is continuous, see The Extreme Value Theorem ( p. 275 )"; hints[4] = "Try f(x)=(x-5)³"; hints[5] = "See Concavity Test ( p. 287 )"; hints[6] = "Try f(x)=(x+4)²"; hints[7] = "

is continuous, see The Intermediate Value Theorem ( p. 127 )"; hints[8] = "f is increasing, so it is one-to-one"; hints[9] = "The product of two increasing functions is an increasing function"; hints[10] = "Try f(x)=x³-6x²+13x"; hints[11] = "Obvious!"; hints[12] = "Try f(x)=x³+x"; hints[13] = "

so 0 is a critical point of

"; hints[14] = "Try f(x)=x³"; hints[15] = "

"; hints[16] = "Try

"; hints[17] = "

is increasing on

. The composition of two increasing functions is an increasing function"; hints[18] = "

, f is increasing on (0,1),

is decreasing and

is decreasing, so

is decreasing on (0,1)"; hints[19] = "

, so

is increasing"; hints[20] = "If

then

"; hints[21] = "

exists, so f is continuous. See The Extreme Value Theorem ( p. 275 )"; hints[22] = "

. See Theorem 6 p. 278"; hints[23] = "

"; hints[24] = "

"; hints[25] = "See The Mean Value Theorem ( p. 283 )"; hints[26] = "Try f(x)=(2x-3)²"; hints[27] = "f has a minimum on [1,2] ( The Extremal Value Theorem ) but does not have a minimum at x=1 and at x=2 ( because f(c)<f(1), f(c)<f(2) )"; hints[28] = "Take f(x)=-(2x+3)²"; hints[29] = "Take f(x)=x³"; hints[30] = "Take f(x)=x³-9x²+26x-24"; hints[31] = "See the second derivative test on p. 287"; hints[32] = "Try f(x)=(x-3)²"; hints[33] = " ( use the limit definition of derivative )"; hints[34] = "

"; hints[35] = "

"; hints[36] = "

"; hints[37] = "Take f(x)=x², g(x)=2x²" hints[38] = "Try f(x)=(x-1)², g(x)=e^-x"; hints[39] = "Try f(x)=x², g(x)=x²"; hints[40] = "f is increasing"; hints[41] = "Try f(x)=(x-1)²"; hints[42] = "If f is decreasing then

"; hints[43] = "Try f(x)=x"; hints[44] = "

"; hints[45] = "

if f(x)<0"; hints[46] = "See the definition of an inflection point ( p. 287 )"; hints[47] = "Try f(x)=x³-x"; hints[48] = "See the definition of concavity ( p. 286 )"; hints[49] = "Take

"; hints[50] = "See the second derivative test ( p. 287 )"; hints[51] = "f is concave down"; hints[52] = "Try f(x)=-x²"; hints[53] = "

!"; probdat[1] = "

is continuous on [2,6] and

for x<3

for x>3

"; adat[1] = "f has a critical point at x=3"; bdat[1] = "f is increasing on (2,3)"; cdat[1] = "f has a maximum on [2,6]"; ddat[1] = "f is concave upward on (2,6)"; probdat[2] = "

for all

"; adat[2] = "f is concave upward on [-3,4]"; bdat[2] = "f has a minimum on [-3,4]"; cdat[2] = "f has a relative minimum on [-3,4]"; ddat[2] = "If

then f has a critical point on [-3,4]"; probdat[3] = "

for all

"; adat[3] = "f is one-to-one"; bdat[3] = "g(x)=x²f(x) is increasing on [1,5]"; cdat[3] = "f has no inflection points"; ddat[3] = "f has a maximum at x=5"; probdat[4] = "f is defined on [-1,1],

exists and c=0 is an inflection point of f"; adat[4] = "f has a critical point on [-1,1]"; bdat[4] = "

has a critical point on [-1,1]"; cdat[4] = "

has a critical point on [-1,1]"; ddat[4] = "c=0 is not a critical point of f"; probdat[5] = "Let f be defined in

and let

"; adat[5] = "if

exists, then

exists"; bdat[5] = "if

exists, then

exists"; cdat[5] = "if f(0)=0,

and f is increasing then g is increasing on (0,1)"; ddat[5] = "if f(0)=0,

and f is concave down and increasing on (0,1) then g is concave down on (0,1)"; probdat[6] = "f is defined on [1,2], one-to-one and

for all

"; adat[6] = "if f is concave down then f ^-1 is concave up"; bdat[6] = "if f is increasing then f ^-1 is decreasing"; cdat[6] = "f does not have a maximum on [1,2]"; ddat[6] = "f does not have a local minimum on (1,2)"; probdat[7] = "f is an odd function and

,

exist"; adat[7] = "

is an even function"; bdat[7] = "

is an odd function"; cdat[7] = "

can be an even nonzero function"; ddat[7] = "

can be an odd nonzero function"; probdat[8] = "f is differentiable on [1,2] and f(1)=f(2)"; adat[8] = "f has a critical point on (1,2)"; bdat[8] = "f has a maximum on (1,2)"; cdat[8] = "if there exists

such that f(c)<f(2) then f has a minimum on (1,2)"; ddat[8] = "f cannot have a maximum on (1,2)"; probdat[9] = "f is defined on [2,4] and

is continuous"; adat[9] = "if

then f has a local minimum at x=3"; bdat[9] = "if

then f has a critical point at x=3"; cdat[9] = "if

then f cannot have a local minimum at x=3"; ddat[9] = "if

then f has an inflection point at x=3"; probdat[10] = "

"; adat[10] = "

exists and 0 is a critical point of f"; bdat[10] = "

does not exist"; cdat[10] = "0 is an inflection point of f"; ddat[10] = "

does not exist"; probdat[11] = "f, g are defined on [-1,3] and

,

for all

"; adat[11] = "f+g is concave upward on (-1,3)"; bdat[11] = "f-g is concave upward on (-1,3)"; cdat[11] = "fg is concave upward on (-1,3)"; ddat[11] = "fg does not have a critical point on (-1,3)"; probdat[12] = "f is defined on [0,1] and

exists on (0,1)"; adat[12] = "if f(0)=0 then

for some

"; bdat[12] = "if

for all

then f has an absolute minimum at x=0"; cdat[12] = "if

for all

then f is increasing"; ddat[12] = "if f is decreasing on (0,1) then

on (0,1)"; probdat[13] = "f is increasing on [-3,3] and

exists on (-3,3)"; adat[13] = "g(x)=f²(x) is increasing on (-3,3)"; bdat[13] = "h(x)=f³(x) is increasing on (-3,3)"; cdat[13] = "if f(x)<0 for -3<x<0 then g(x)=f²(x) is decreasing on (-3,0)"; ddat[13] = "if f(x)<0 for 0<x<3 then h(x)=f³(x) is decreasing on (0,3)"; probdat[14] = "

is increasing on (-1,0) and decreasing on (0,1)"; adat[14] = "0 is an inflection point of f"; bdat[14] = "0 is a critical point of f"; cdat[14] = "f is concave up on (-1,0)"; ddat[14] = "f is decreasing on (0,1)"; probdat[15] = "

for all

"; adat[15] = "if

for some -4<c<4 then f does not need to have a maximum at c"; bdat[15] = "f has no inflection points on (-4,4)"; cdat[15] = "f is increasing on (-4,4)"; ddat[15] = "

is decreasing on (-4,4)"; 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=no,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 answers:

"); 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 answers:

"); 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(); }