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Solved: Comparing the Midpoint and Trapezoid Rules Compare

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett ISBN: 9780321570567 2

Solution for problem 43E Chapter 7.6

Calculus: Early Transcendentals | 1st Edition

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Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

Calculus: Early Transcendentals | 1st Edition

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Problem 43E

Problem 43E

Comparing the Midpoint and Trapezoid Rules

Compare the errors in the Midpoint and Trapezoid Rules with n =4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

Step-by-Step Solution:
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Week 7 Abnormal Psychology Notes Symptoms of Manic Episodes  Inflated self-esteem or grandiosity  Decreased need for sleep  More talkative than usual or pressure to keep talking  Subjective experience that one’s thoughts are racing or flight of ideas  Distractibility  In Bipolar disorders, manic episodes tend to be shorter than depressive episodes Specifiers of Bipolar Disorders  Rapidly cycling bipolar disorder o 4+ severe mood episodes in a single year o Usually an individual will experience less than one full cycle in a year  Mixed State o Sxs of mania and depression that occur at the same time o Very dangerous DSM-5 Controversies  Removal of bereavement exclusion for Major Depressive Disorder o Bereavement exclusion as the inability to diagnose someone with MDD who just experienced a death of a loved one  New: Disruptive Mood Dysregulation Disorder o Diagnosis added to address children who may be Bipolar  New: Premenstrual Dysphoric Disorder o Diagnosis for extreme PMS symptoms o Use antidepressants to treat Epidemiology  Major Depression is one of the most common single psychiatric disorders worldwide o 19% lifetime pre. for U.S. adults o Higher rates in women and in low-income or oppressed populations  Bipolar Disorders are less common o 4% lifetime pre. for U.S. adults o Equally common in men and women o Higher rates in low-income populations Developmental Factors  Depression can occur at any age, even infancy o Sxs differ by age, median age of onset is 26 years old o In childhood, gender ratio is equal, but in adolescence, rates for girls increases Week 7 Abnormal Psychology Notes  Bipolar disorders can affect all ages, but diagnosis is controversial in young children o In older adults, mania and depression often result from medical illnesses Etiology  Stress factors o Depressive Disorders include stress from environment and life events  Exogenous (caused by external stress) vs. Endogenous (caused by internal stress) o Bipolar Disorders cannot conclude  Possible indication to stress  Genetic Factors o Depressive Disorders: genetics play a predisposing role in some forms  Heritability: 30-40%  Twins: MZ: 46% DZ: 20% o Bipolar Disorders: strong genetic basis  Heritability: 60-80%  Twins: MZ: 40% DZ: 5%  Biochemical Factors o Depressive Disorders have abnormal NT function of monoamines (a class of NTs that deal with mood)  Permissive theory: low levels of serotonin disrupts function of other NTs that affect mood  Low serotonin + low NE = Depression  Low serotonin + high NE = Mania  Endocrine system (hormone release): people with depression have high levels of cortisol and abnormal levels of melatonin  Protein deficiencies within neurons (BDNF=Brain-derived neurotrophic factor)  Exposure to stress = a decrease in BDNF which = a decrease in hippocampus size  Exercise increase BDNF o Bipolar Disorders have low NT function and abnormal ion activity of neuronal membranes  Ions are needed to send messages down the axon of a neuron, sometimes they fire too easily or not enough  Brian Anatomy, Circuits, and Structures o Depressive Disorders  Prefrontal cortex  hippocampus  amygdala  Brodmann area 25  This circuit is filled with serotonin transporters  The Brodmann area 25 is much more active in depressed people Week 7 Abnormal Psychology Notes  The hippocampus stops making new cells when you’re depressed and decreases in size o Bipolar disorders vary, there are several abnormal structures, such as basal ganglia and cerebellum, but roles are unclear  Immune System Factors o Depressive Disorders  Immune system dysregulation is due to stress, this may be a cause OR result of depression, it is a correlational finding only  Psychodynamic Factors o Depressive Disorders  Difference between grief and depressions (Freud)  Centrality of loss – anaclitic depression (conflicted feelings)  Anger turned inward; self-punitiveness  Problematic relationships  Impaired attachment (childhood needs were not met)  Abandonment and parental criticism  Children and a parents depression  Cognitive Factors o Depressive Disorders  Negative thinking  Maladaptive attitudes, cognitive distortions or errors in thinking  Beck’s negative cognitive triangle: self, world and future  Negative thoughts are automatic  Learned Helplessness  Lack of control over rewards/punishments lead to depression o Giving up because you feel you have no control  Seligman study with the dogs in the shocking box to prove learned helplessness  Attribution-Helplessness Theory: internal, global, stable attributions for negative events leads to depression  Sociocultural Factors o Depressive Disorders are linked with lack of social supports and isolation  Strong link between gender and depression (women are more likely to be diagnosed)  Cultural background  Hispanics and Blacks are 50% more likely to develop depression than white people  Nonwestern countries associate depression with physical factors  Western countries focus on the cognitive part Week 7 Abnormal Psychology Notes Gender and Depression  Females are twice as likely to be diagnosed with depression as males  “Role/Life Stress” theory: demanding but undervalued roles; expected to fulfill several roles simultaneously (career, housekeeping, childcare); lower pay and harsher evaluations for same job as men  Rumination theory: socialized to internalize distress while males externalize it  “Lack of Control” theory: far more likely to experience forms of victimization associated with depression (rape, spousal abuse, childhood sexual abuse) and socioeconomic disadvantage Treatment for Mood Disorders  Depressive Disorders o Approximately 85% of people with unipolar depression recover, some without treatment o Big range of treatment options like many types of therapy and medication  Bipolar Disorders o More than 60% of patients with mania improve with medication and monitoring o Medication is the primary treatment  Lithium and psychotherapy  Divided into Psychological and Biological treatments Psychological Treatments  Psychodynamic Therapy o Depressive Disorders  Efficacy debated  Focus on loss, anger, problematic childhood experiences/relationships, repetitive life patterns  Can be short or long term therapy o Bipolar Disorders  Supportive: identifying emotional triggers for mood swings, supporting healthy coping  Behavioral therapy o Depressive Disorders  Not used by itself anymore, usually combined with cognitive therapy  Reintroduce clients to pleasurable activities; reinforce non-depressive behaviors; improve social skills  Behavioral activations  Very simple structure, activities you have to do that you enjoy  Lasting effects are unclear  Cognitive therapy Week 7 Abnormal Psychology Notes o Depressive Disorders  Often combined with behavioral therapy (CBT)  Four phases (usually fewer than 20 sessions)  Increasing activities and elevating mood  Challenging automatic thoughts  Identifying negative thinking and biases  Changing primary attitudes  Acceptance and commitment therapy  Accepting negative thoughts and recognizing that, and living with them to move on o Bipolar Disorders  Altering belief of self as helpless  Interpersonal Psychotherapy (IPT) o Depressive Disorders  Focuses on problematic relationships or coping with loss of relationships  Structure of CBT but content of psychodynamic theory; recognizes key role of social stressors o Bipolar Disorders  Interpersonal social rhythm therapy: routines (daily activities and mood watching)  Others o Feminist therapy o Family therapy o Couples therapy o Culturally-sensitive therapy

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Chapter 7.6, Problem 43E is Solved
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Textbook: Calculus: Early Transcendentals
Edition: 1
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
ISBN: 9780321570567

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Solved: Comparing the Midpoint and Trapezoid Rules Compare