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If a function \\(f\\) is defined on a closed interval \\([a,b]\\), then \\(f\\) attains an absolute maximum value \\(f(c)\\) and an absolute minimum value \\(f(d)\\) at some numbers \\(c\\) and \\(d\\) in \\([a,b]\\). <\/p>\n"},"subContentId":"5e090509-7bcd-43ff-a554-aa61dfd50581","metadata":{"contentType":"True\/False Question","license":"U","title":"True\/False Question"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

T\/F<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":145.065,"to":155.065},"libraryTitle":"Multiple Choice","action":{"library":"H5P.MultiChoice 1.16","params":{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

The area always increases.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

The area will first increase then decrease.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

The area will eventaully become 0 again.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":true,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

Can you predict once the width of the rectagle gets shorter than \\(\\frac{1}{2}\\)m, and the height of the rectangle gets taller than 0, what happens to the area of the rectangle?<\/p>\n"},"subContentId":"f26cb7ae-10fa-4d09-a561-8a02068478c7","metadata":{"contentType":"Multiple Choice","license":"U","title":"Multiple Choice: How would the area of the rectangle change?"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

MCQ<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":274.425,"to":284.425},"libraryTitle":"Multiple Choice","action":{"library":"H5P.MultiChoice 1.16","params":{"media":{"disableImageZooming":false},"answers":[{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

\\(x(\\frac{1}{2} -x)\\)<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

\\(y(\\frac{1}{2} -y)\\)<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

\\(\\frac{1}{2}x - x^2\\)<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

\\(\\frac{1}{2}y - y^2\\)<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":true,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

We can rewrite the function \\(s\\) as:<\/p>\n"},"subContentId":"ece6090c-487d-4a3a-adc6-64827cdb8528","metadata":{"contentType":"Multiple Choice","license":"U","title":"Multiple Choice: Choose the right option(s)."}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

What is function \\(s?\\)<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":525.419,"to":535.419},"libraryTitle":"True\/False Question","action":{"library":"H5P.TrueFalse 1.8","params":{"media":{"disableImageZooming":false},"correct":"false","behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"feedbackOnWrong":"Think about function such as \\(f(x) = x^3\\). At \\(x =0,\\) we have \\(f^\\prime (0) = 0\\) but the function doesn't have a maximum nor a minimum at \\(x =0.\\)"},"l10n":{"trueText":"True","falseText":"False","score":"You got @score of @total points","checkAnswer":"Check","showSolutionButton":"Show solution","tryAgain":"Retry","wrongAnswerMessage":"Wrong answer","correctAnswerMessage":"Correct answer","scoreBarLabel":"You got :num out of :total points","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

When \\(f^\\prime (c) =0\\), the function \\(f\\) has a maximum or minimum at \\(c\\).<\/p>\n"},"subContentId":"1f446bfb-bf27-4ab7-930b-4831d347c6f2","metadata":{"contentType":"True\/False Question","license":"U","title":"Untitled True\/False Question"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

Fermat's Theorem<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":572.594,"to":582.594},"libraryTitle":"Drag the Words","action":{"library":"H5P.DragText 1.10","params":{"taskDescription":"Drag the words into the correct boxes","overallFeedback":[{"from":0,"to":100}],"checkAnswer":"Check","tryAgain":"Retry","showSolution":"Show solution","dropZoneIndex":"Drop Zone @index.","empty":"Drop Zone @index is empty.","contains":"Drop Zone @index contains draggable @draggable.","ariaDraggableIndex":"@index of @count draggables.","tipLabel":"Show tip","correctText":"Correct!","incorrectText":"Incorrect!","resetDropTitle":"Reset drop","resetDropDescription":"Are you sure you want to reset this drop zone?","grabbed":"Draggable is grabbed.","cancelledDragging":"Cancelled dragging.","correctAnswer":"Correct answer:","feedbackHeader":"Feedback","behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"instantFeedback":false},"scoreBarLabel":"You got :num out of :total points","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again.","textField":"Suppose \\(f^{\\prime \\prime}\\) is continuous near \\(c\\). \nIf \\(f^\\prime (c) = 0\\) and \\(f^{\\prime \\prime}(c) >0,\\) then \\(f\\) has a *local minimum* at \\(c\\). If \\(f^\\prime (c) = 0\\) and \\(f^{\\prime \\prime}(c) <0,\\) then \\(f\\) has a *local maximum* at \\(c\\)."},"subContentId":"3e58e2ee-77f4-4393-9e8a-d77fb704cccc","metadata":{"contentType":"Drag the Words","license":"U","title":"Untitled Drag the Words"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

2nd derivative test<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":772.282,"to":782.282},"libraryTitle":"Multiple Choice","action":{"library":"H5P.MultiChoice 1.16","params":{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

It's decreasing.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

It's increasing.<\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

It's decreasing then increasing.<\/div>\n"},{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

It's increasing then decreasing.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":true,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

If you are going to make a guess about how the distance between P and a point on the straight line is going to change, what would it be? Assume the point on the straight line \\(y=2x\\) will travel from the rightmost of x-axis, to the leftmost of x-axis.<\/p>\n"},"subContentId":"28962b66-ea7c-4358-85ff-0edf4cae64fb","metadata":{"contentType":"Multiple Choice","license":"U","title":"Untitled Multiple Choice"}},"pause":false,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

MCQ<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":862.764,"to":872.764},"libraryTitle":"True\/False Question","action":{"library":"H5P.TrueFalse 1.8","params":{"media":{"disableImageZooming":false},"correct":"true","behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false},"l10n":{"trueText":"True","falseText":"False","score":"You got @score of @total points","checkAnswer":"Check","showSolutionButton":"Show solution","tryAgain":"Retry","wrongAnswerMessage":"Wrong answer","correctAnswerMessage":"Correct answer","scoreBarLabel":"You got :num out of :total points","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

The function \\(D(x,y) = \\sqrt{x^2+ (y-1)^2}.\\)<\/p>\n"},"subContentId":"19b72096-27bc-408a-87be-8a4aea6b483b","metadata":{"contentType":"True\/False Question","license":"U","title":"Untitled True\/False Question"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

T\/F<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":1017.932,"to":1027.932},"libraryTitle":"Multiple Choice","action":{"library":"H5P.MultiChoice 1.16","params":{"media":{"disableImageZooming":false},"answers":[{"correct":false,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

Yes we do since we have a fraction, and when the denominator of a fraction is equal to 0, the fraction is not defined. <\/div>\n"},{"correct":true,"tipsAndFeedback":{"tip":"","chosenFeedback":"","notChosenFeedback":""},"text":"

No we don't: even though we have a fraction, its denominator is never 0 in this example.<\/div>\n"}],"overallFeedback":[{"from":0,"to":100}],"behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"type":"auto","singlePoint":false,"randomAnswers":true,"showSolutionsRequiresInput":true,"confirmCheckDialog":false,"confirmRetryDialog":false,"autoCheck":false,"passPercentage":100,"showScorePoints":true},"UI":{"checkAnswerButton":"Check","showSolutionButton":"Show solution","tryAgainButton":"Retry","tipsLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","tipAvailable":"Tip available","feedbackAvailable":"Feedback available","readFeedback":"Read feedback","wrongAnswer":"Wrong answer","correctAnswer":"Correct answer","shouldCheck":"Should have been checked","shouldNotCheck":"Should not have been checked","noInput":"Please answer before viewing the solution","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again."},"confirmCheck":{"header":"Finish ?","body":"Are you sure you wish to finish ?","cancelLabel":"Cancel","confirmLabel":"Finish"},"confirmRetry":{"header":"Retry ?","body":"Are you sure you wish to retry ?","cancelLabel":"Cancel","confirmLabel":"Confirm"},"question":"

We know that a critical number of a function \\(f\\) is a number \\(c\\) in the domain of \\(f\\) such that either \\(f^\\prime (c) =0\\) or \\(f^\\prime(c)\\) does not exist. In this example, do we need to worry about the case when the derivative does not exist?<\/p>\n"},"subContentId":"5814f8da-48bb-4feb-b04e-ffb2fafe5e2b","metadata":{"contentType":"Multiple Choice","license":"U","title":"Untitled Multiple Choice"}},"pause":false,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":"

MCQ<\/p>\n"},{"x":47.81659388646288,"y":46.10894941634241,"width":10,"height":10,"duration":{"from":1098.549,"to":1108.549},"libraryTitle":"Drag the Words","action":{"library":"H5P.DragText 1.10","params":{"taskDescription":"Drag the words into the correct boxes","overallFeedback":[{"from":0,"to":100}],"checkAnswer":"Check","tryAgain":"Retry","showSolution":"Show solution","dropZoneIndex":"Drop Zone @index.","empty":"Drop Zone @index is empty.","contains":"Drop Zone @index contains draggable @draggable.","ariaDraggableIndex":"@index of @count draggables.","tipLabel":"Show tip","correctText":"Correct!","incorrectText":"Incorrect!","resetDropTitle":"Reset drop","resetDropDescription":"Are you sure you want to reset this drop zone?","grabbed":"Draggable is grabbed.","cancelledDragging":"Cancelled dragging.","correctAnswer":"Correct answer:","feedbackHeader":"Feedback","behaviour":{"enableRetry":true,"enableSolutionsButton":true,"enableCheckButton":true,"instantFeedback":false},"scoreBarLabel":"You got :num out of :total points","a11yCheck":"Check the answers. The responses will be marked as correct, incorrect, or unanswered.","a11yShowSolution":"Show the solution. The task will be marked with its correct solution.","a11yRetry":"Retry the task. Reset all responses and start the task over again.","textField":"The First Derivative Test: Suppose that \\(c\\) is a critical number of a continuous function \n\\(f\\).\u00a0\n\nIf \\(f^\\prime\\) changes from positive to negative at \\(c\\), then \\(f\\) has a *local maximum* at \n\\(c\\).\nIf \\(f^\\prime\\) changes from negative to positive at \\(c\\), then \\(f\\) has a *local minimum* at \n\\(c\\).\nIf \\(f^\\prime\\) is positive to the left and right of \\(c\\), or negative to the left and right of \\(c\\), then \\(f\\) has *no local maximum or minimum* at \\(c\\)."},"subContentId":"f4f8591d-3bd4-4425-876a-9cf0735a1fbe","metadata":{"contentType":"Drag the Words","license":"U","title":"Untitled Drag the Words"}},"pause":true,"displayType":"button","buttonOnMobile":false,"adaptivity":{"correct":{"allowOptOut":false,"message":""},"wrong":{"allowOptOut":false,"message":""},"requireCompletion":false},"label":""}]},"summary":{"task":{"library":"H5P.Summary 1.10","params":{"intro":"Choose the correct statement.","summaries":[{"subContentId":"adbdbf16-93a1-4443-aa12-bab3864a4a37","tip":""}],"overallFeedback":[{"from":0,"to":100}],"solvedLabel":"Progress:","scoreLabel":"Wrong answers:","resultLabel":"Your result","labelCorrect":"Correct.","labelIncorrect":"Incorrect! Please try again.","alternativeIncorrectLabel":"Incorrect","labelCorrectAnswers":"Correct answers.","tipButtonLabel":"Show tip","scoreBarLabel":"You got :num out of :total points","progressText":"Progress :num of :total"},"subContentId":"508412e8-10a6-45da-be98-0d7df2881bbb","metadata":{"contentType":"Summary","license":"U","title":"Untitled Summary"}},"displayAt":3}},"override":{"autoplay":false,"loop":false,"showBookmarksmenuOnLoad":false,"showRewind10":false,"deactivateSound":false,"preventSkippingMode":"none"},"l10n":{"interaction":"Interaction","play":"Play","pause":"Pause","mute":"Mute, currently unmuted","unmute":"Unmute, currently muted","quality":"Video Quality","captions":"Captions","close":"Close","fullscreen":"Fullscreen","exitFullscreen":"Exit Fullscreen","summary":"Open summary dialog","bookmarks":"Bookmarks","endscreen":"Submit screen","defaultAdaptivitySeekLabel":"Continue","continueWithVideo":"Continue with video","playbackRate":"Playback Rate","rewind10":"Rewind 10 Seconds","navDisabled":"Navigation is disabled","sndDisabled":"Sound is disabled","requiresCompletionWarning":"You need to answer all the questions correctly before continuing.","back":"Back","hours":"Hours","minutes":"Minutes","seconds":"Seconds","currentTime":"Current time:","totalTime":"Total time:","singleInteractionAnnouncement":"Interaction appeared:","multipleInteractionsAnnouncement":"Multiple interactions appeared.","videoPausedAnnouncement":"Video is paused","content":"Content","answered":"@answered answered","endcardTitle":"@answered Question(s) answered","endcardInformation":"You have answered @answered questions, click below to submit your answers.","endcardInformationNoAnswers":"You have not answered any questions.","endcardInformationMustHaveAnswer":"You have to answer at least one question before you can submit your answers.","endcardSubmitButton":"Submit Answers","endcardSubmitMessage":"Your answers have been submitted!","endcardTableRowAnswered":"Answered questions","endcardTableRowScore":"Score","endcardAnsweredScore":"answered","endCardTableRowSummaryWithScore":"You got @score out of @total points for the @question that appeared after @minutes minutes and @seconds seconds.","endCardTableRowSummaryWithoutScore":"You have answered the @question that appeared after @minutes minutes and @seconds seconds."}}

Optimization Lecture

Xinli Wang