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Answer: In each of 13 through 18:(a) Sketch the graph of the given function for three

Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce ISBN: 9780470458310 168

Solution for problem 16 Chapter 10.2

Elementary Differential Equations and Boundary Value Problems | 10th Edition

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Elementary Differential Equations and Boundary Value Problems | 10th Edition | ISBN: 9780470458310 | Authors: William E. Boyce

Elementary Differential Equations and Boundary Value Problems | 10th Edition

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Problem 16

In each of 13 through 18:(a) Sketch the graph of the given function for three periods.(b) Find the Fourier series for the given function.f(x) =x + 1, 1 x < 0,1 x, 0 x < 1; f(x + 2) = f(x)

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Csikszentmihalyi’s Theory of the Unconscious in Creative Thinking  Rejects Freudian primary-process view as “spectacularly implausible”  Wallas’s stages – basis of Csiks (1996) theorizing o “cognitive view” which assumes unconscious thinking involves making connections among ideas based on laws of simple association = random combination of ideas  Conscious = logical, linear o Rejects Freud (1996) but agrees that unconscious is able to make connections that conscious cannot  Uses analogy similar to Poincare’s discussion of aesthetically pleasing combinations being passed to consciousness – when insight occurs o i.e. (1996) Kekule if he stayed awake, rationality would have censored the connection Interviewees’ Opinions on the Unconscious  Csizkszentmihalyi and Sawyer (1995) o Interviewed 100 individuals who claim creative leaps from unconscious processing important in creative work o Described moments of insight as part of 4 stage process – Wallas (1926): preparation, incubation, illumination, verification o Emphasizes individual differences that can occur in incubation and illumination  Long time frame vs. short time frame (two ends of continuum)  Presented vs. discovered problems o Presented: shorter time frame, problem already exists o Discovered problems: long time frame, problem isn’t one people have dealt with before  Great creative insights that result in shifts in the field (revolutionary = paradigm shift)  HOWEVER, Darwin was working on presented problem when he caused the paradigm shift Interaction between Conscious and Unconscious Processes  Csikszentmihalyi and Sawyer (1995)unconscious processing (parallel processor with greater capacity) has greater capacity than conscious processing (serial processor with limited capacity) o Many small entities that can work on many different problems at once o Individual must be able to direct the undirectable subconscious process  useful insight  Many of interviewees claimed to have this ability  Found no evidence of external stimulus whose action led to insight o “welling up from the subconscious” o Poincare’s walking on the buff & boarding plane Data: Content of the Interviews  Csizentmihalyi and Sawyer (1995) o Short time frame - Many respondents structured days to include period of idle time followed by period of hard work (needed for “best” ideas)  Usually during repetitive physical activity  Keep journal to capture ideas o Longer time frame – insight happened during extended period of time away from work  vacation o Elaboration stage – almost all reported that it was necessary  One environmentalist/economist/poet did very little editing afterwards o Conclusion  Creative leap/insight is result of period of incubation, during which info is processed in parallel at an unconscious level  If incubation successful  insight  conscious evaluation and elaboration of the illumination The Question of Subjective Reports  Poincare, Wallas, Hadamard, Koestler, Campbell all driven by logic and all built on existing ideas of eachother o Poincare used logic to explain his own experience o Wallas provided little new evidence beyond Poincare’s o Hadamard added reports from Einstein and others (still self-reports) & used logic to describe own experiences of facial recognition and speech production o Koestler used self reports and incorporated ideas from Freud and Poincare o Campbell used Poincare’s reports as evidence for blind-variation theory (didn’t provide any new data) o Simonton accepted argumens of Poincare & Campbell + Koestler’s ideas (and Freud’s) o Simonton adopted Mednick’s theory (cited Poincare) o Csikszentmihalyi and Sawyer build on Poincare’s theory, filtered through Wallas’s stages Laboratory Investigations of Incubation and Illumination  Search for evidence to verify stages of creative thinking  Search for evidence of illuminations Patrick’s Studies of Stages of Creative Thinking  C. Patrick (1935) o Accomplished poets given picture and asked to write poem about it  C. Patrick (1937) o Accomplished artists asked to draw picture in response to poetry of Milton  BOTH o Research  Performance compared to control group who had not exhibited creative achievement  Carried out task while thinking aloud  Patrick took notes shorthand  Single session  Afterwards, Patrick interviewed participants about usual methods of working (incubation and illumination) o Findings  Most reported that they used methods that they usually used  All took 20 minutes  Patrick divided into quarters (4 stages) o 1. Made the most shifts from idea to idea – Preparation o 2. Recurrence of previously rejected idea most frequent – Incubation o 3. Project given general theme or shape that remained – illumination o 4. Highest frequency of revisions - Verification o Questions  1 session = individual thinking about the creative project the whole time = no impasse/idle time/incubation o Take aways  Artists reported period of incubation  Think about ideas off and on when not formally working on a project  Illumination could be the result o Unverified responses, no other evidence Eindhoven and Vinacke’s Study of Stages in Creative Thinking  Eindhoven and Vinacke (1952) o Research:  Extended Patrick’s research to try to overcome limitations  13 Painters asked to create picture in response to poem  Up to 4 sessions over a week’s time  Almost all took 2+ sessions  More than half 3+ (3/4 non artists)  Some artists visits lasted across 2 weeks, one took several months between 2 visits  No time limit on visits, 1 hr suggested  Participants asked to keep diary of experiences and ideas relevant to project o Findings:  Multiple sessions used to carry out project  Artists produced many more sketches than non-artists  Artists tended to produce new sketches only during 1 session (small) & concentrated on one basis for final work  Non artists produced new sketches over several sessions  Produced larger works throughout sessions  Both artists and non-artists spent early time laying out general aspects of picture then later, detail  Like Guernica  Only 7 participants kept diaries (5/7 artists)  Post study interviews showed they thought about project while away from studio o Conclusions:  Artists and non-artists behaved very differently - same process doesn’t occur in all individuals when faced with situation requiring creative thought  Stages described by Poincare & Wallas could not be isolated as separable entities  Stages blend into eachother in complex ways  Hard to ID specific point where idea came into consciousness  Illumination is process not a stage Attempts to Demonstrate Incubation in the Experimental Laboratory  Table 8.3A (pg. 420-421) o If break results in better performance = evidence for incubation during idle time  Assuming they didn’t think of problem during break (if so, no incubation + extra time to work)  Leave lab  Do distractor task in lab o Nothing to do with problem o Problems of same sort o Does taking a break facilitate problem solving  Mixed reviews  Flaws in design of studies o C. Patrick (1938)  Research: asked individuals to propose scientific methods to investigate effects of heredity and environment on humans  Control group worked continuously  Incubation group given diary and told to return in 2-3 weeks o Didn’t take a break – given more time to work o Olton and Johnson (1976)  Research: examined possible effects of several different sorts of activities during the incubation period  Unstructured free time  listening to set-breaking instructions  being shown geometric forms that were analogous to solution  Results: no incubation, solved problem at same rate as control  Attempt to replicate Dreistadt (1969) o Olton and Johnson (1979)  Research: Gave experienced chess players 1 hour to work on a problem chess master deemed difficult  If didn’t solve in hour, took 2 hour break o Could do anything they wanted except think about chess  Findings: only 1 Aha! Moment during break; didn’t find incubation  Conclusion: “creative worrying” – actively thinking about problem when not supposed to = incubation o Browne and Cruse (1988, Experiment 2)  Research: Used Farm problem and examined 3 distractor activities  Analogical hint group - drew geometric forms Relaxation group – listened to music and given relaxation instructions  Difficult mental work – memorizing a passage  Findings: incubation effects found for analogical hint and relaxation groups; no effects for difficult mental work  Conclusion: positive effects due to participants’ working on problem when they weren’t supposed to be thinking about it (conscious thinking)  Maybe this is actually what occurs during “incubation” o A.S. Patrick (1986)  Research: study using verbal insight problems  30 problems, then 5 randomly chosen from unsolved problems used to test for incubation o One cycled through, 2 minutes at a time until 8 minutes up o One group cycled through but spent additional 5 minutes talking with experimenter about unrelated activities first o One group cycled and carried out difficult mental task for 5 minutes in between  Impasse, several problems used to test incubation, experimental controls made it difficult for people to think about problems during incubation periods  Findings: only incubation in problems plus mental rotation with high-ability participants o Segal (2004)  Research: Used geometry problem  Began incubation only after participant indicated they had reached impasse  Incubation periods 4-12 minutes  Distractor activities easy or difficult  Findings: incubation with hard task for both short and long incubation periods; less strong effect for easy task with only the short interval resulting in increased performance  No strong pattern of incubation Laboratory Studies of Incubation: Conclusions  Lab studies sterile and far removed from real-life situations in which incubation results in illumination  At least one study produced negative results o Olton and Johnson (1976) Chess study  Almost never report Aha! Experiences o Only show whether taking break helps solve problem; not parallel to Poincare’s experience Evidence for Incubation and Illumination: A Critique  Postulation of unconscious processing in creative thinking based mainly on anecdotal reports  Table 8.4 (pg. 429) Questions about Poincare’s Self-Reports  Public discussion of discoveries presented 30 years after occurrence o Brief occurrences hard to remember after long period of time  No experience in behavioral sciences o No training on observing and reporting behavior  Poincare was conscious during sleepless night Poincare’s Logical Analysis  Based on several assumptions about thinking process in general and creative thinking in particular that aren’t the only possible assumptions o Poincare’s theory of thinking is a bottom-up view  Works by combining basic elements/ideas without any planning o Top Down process is possible  Goal of thinker plays role in determining how ideas combined  Combinations not tried out because never considered o i.e. Watson and Crick decision to deal with helixes limited breadth of conscious search o All research based on Poincare is questionable too Modern Views of Unconscious Processing  Csikszentmihalyi and Sawyer (1995) participant said ideas come to her while gardening, no evidence of unconscious processing o Conscious o Could be thinking while gardening  Physicist-mathematician conscious on bus ride to California and was probably consciously thinking about the problem during this “eureka!” experience  Banker’s memo from the beach – he was thinking about the business and wrote a memo (conscious act) Illumination without Unconscious Processing Selective Forgetting  Woodworth (1938) o Argued that initial attempts to solve problem were in wrong direction and that nothing positive, conscious or unconscious occurs during break; break just allows time for change of cues eliciting the mode of dealing with the problem  Misdirecting factors selectively forgotten during interval  Smith (1995) o Unsuccessful initial problem solving attempts  mental ruts that interfere with thinking of new approaches  Unsuccessful cues must be forgotten before anything different generated  Research: People given misdirecting cue then presented problem again after break and also asked to remember cues  Findings: o longer breaks = higher chance of solving problem o longer breaks = harder to remember misdirecting cues Selective Forgetting and Illumination: Critique  No spontaneous solution o During Poincare’s walk on beach; problem not re-presented to him (like in Smith’s experiment) – spontaneously came to mind Failure Indices: The Opportunistic-Assimilation Model  Seifert and colleagues (1995) o Theory: Opportunistic-assimilation model of illumination and insight  Individual assimilates relevant environmental events to stored memory of Unsolved problem + failure index/gap o Research:  Phase 1 – participants given general-information questions  Phase 2 – participants do word-recognition task (some words = answers to phase 1 questions)  Phase 3 – next day, revisit some of Phase 1 questions and some new ones o Findings:  Phase 2 facilitated solution of phase 1 problems when re- presented in phase 3 o Conclusions:  Incubation doesn’t play role in problem solving  Break helped lead to accidental encounter of relevant stimuli Illumination as the Result of Spontaneous Retrieval of a Problem: Critique of the Failure Index Hypothesis  Not relevant to failure index model or Poincare’s reports of his experiences (Table 8.6C pg. 437) o Rely on relevant environmental cue o Still have to re-present problem  Environmental cue doesn’t cause spontaneous retrieval of problem  Dodds and colleagues (2002) o Research: used three-word verbal insight problems to examine influence of environmental cues on solution of previously unsolved problem  20 problems, 30 seconds for each nd  Told that they would have 2 attempt at problems  Next, given activity  Told that it would help with solving problems o Either answers to earlier problems, words semantically related, or unrelated words o Finding:  Presentation of solution words & related words only helpful when participants knew that the activity may help with problem solving o Conclusions:  Doesn’t support opportunistic-assimilation view  Doesn’t support spontaneous retrieval/Aha! experience  Anolli and colleagues (2001) o Research:  Problem similar to Radiation problem was target Participants presented with serious of problems that required them to retain info presented in stories; analogous problem to target problem also presented o Reminder group: reminded that they needed to answer question about base story read earlier o Hint group: hinted based info could suggest solution for target problem o Reminder + Hint Group: both o Findings:  Solution of target problem only when participants heard hint that base info relevant o Conclusions:  Contradict opportunistic-assimilation model  Doesn’t support spontaneous retrieval  Christensen and Schunn (2005) o Research:  Cues for earlier unsolved problems presented as participant worked through series of problems; (period of time passed/worked on few irrelevant problems before cues presented)  When given analogous base problem and solution, asked to rate difficulty of the problem Could return to any unsolved problem whenever they wished o Findings:  Support for opportunistic-assimilation theory  Presentation of cue analogous to unsolved problem  retrieval of that problem and its solution o Possible that Poincare got environmental cue during walk as well o Critique:  Cues presented in same context  Poincare’s illumination occurred in context very different from his work context Creative Worrying Conscious Thinking during a Break  Olton (1979) o Proposes that no unconscious processes occur; person consciously thinking about/working on problem during “incubation” period  Reported by participants in Eindhoven and Vinacke (1952) o Critique:  No evidence of this work in Poincare Degrees of Creative Worrying: Brief Conscious Interludes  Weisberg o Brief conscious interlude  people don’t intentionally return to thought about the problem; your mind just finds itself thinking about it randomly  It is hard to publicly project conscious ideas that are brief, unintentional  May or may not be relevant  If not relevant, hard to report except immediately after  If problem solved from brief conscious interlude, may not remember where solution came from  Done so quickly that conscious post responding report is limited

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Textbook: Elementary Differential Equations and Boundary Value Problems
Edition: 10
Author: William E. Boyce
ISBN: 9780470458310

The full step-by-step solution to problem: 16 from chapter: 10.2 was answered by , our top Math solution expert on 12/23/17, 04:36PM. This full solution covers the following key subjects: . This expansive textbook survival guide covers 76 chapters, and 2039 solutions. Since the solution to 16 from 10.2 chapter was answered, more than 250 students have viewed the full step-by-step answer. The answer to “In each of 13 through 18:(a) Sketch the graph of the given function for three periods.(b) Find the Fourier series for the given function.f(x) =x + 1, 1 x < 0,1 x, 0 x < 1; f(x + 2) = f(x)” is broken down into a number of easy to follow steps, and 41 words. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10.

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Answer: In each of 13 through 18:(a) Sketch the graph of the given function for three