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Find the absolute minimum and maximum values of the

Vector Calculus | 6th Edition | ISBN: 9781429215084 | Authors: Jerrold E. Marsden; Anthony Tromba ISBN: 9781429215084 87

Solution for problem 4 Chapter 3

Vector Calculus | 6th Edition

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Vector Calculus | 6th Edition | ISBN: 9781429215084 | Authors: Jerrold E. Marsden; Anthony Tromba

Vector Calculus | 6th Edition

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3
Problem 4

Find the absolute minimum and maximum values of the function f (x, y) = x2 + 3xy + y2 + 5 on the unit disc D = {(x, y) | x2 + y2 1}.

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Step 1 of 3

Exam 2 Study Guide Hormones ­ Gradual, slow, long distance, diffuse, can depend on time, graded effect ­ Usually temporary effect, but not always ­ Cushing’s disease- from excess glucocorticoids (from steroids) Endocrine pathway 1) Hypothalamus (secretes releasing hormone/TRH) 2) Pituitary (anterior secretes tropic hormones/TSH) o Posterior – Vasopressin and oxytocin/Anterior- All other hormones 3) Thyroid (secretes thyroid hormone) a. Act on target cells or 1 or 2 to inhibit (negative feedback) b. Thyroid medicine  increased thyroid hormone i. Body compensates TSH to opposite of thyroid level 1. High serum thyroid = low TSH c. Needs iodine i. Goiter – swollen gland from iodine deficiency d. Thyroid deficiency at birth can cause microcephaly Gender ­ Not a dichotomy, a continuum ­ 7 levels of determination o 1) Chromosomal sex (SRY on Y chrom. makes testes) o 2) Gonadal sex (whether you have testes or not) o 3/4) Internal and external organs o 5) Brain sex (brain structures)  Women have a larger corpus callosum (cognitive skills)  Women have greater gray matter % in dorsolateral prefrontal cortex (planning behavior) and superior temporal gyrus (language)  Males have greater asymmetry than females  Thicker/larger R hemisphere  Males are better connected front-to-back (focusing)  Women are better connected left-to-right (multitasking) o 6/7) Gender identity; gender preference ­ Organizing effects – occur prenatally or shortly after birth (testosterone) ­ Activating effects – occur at any time; reversible ­ Behavior differences – mostly overlap Sexual orientation ­ Biological hypothesis (involves early changes, birth order, genetic and hormonal influence) o Gay and straight males have the same testosterone levels but prenatal exposure to testosterone may have an effect ­ Brain involvement o Women and gay men (vs heterosexual men) have a…  Larger anterior commissure (white matter – communication between L and R hemispheres) – related to visual/spatial ability  Smaller INH3 (may be testosterone related)  Larger suprachiasmatic nucleus (SCN) Sex ­ Involves hormonal control and specific brain areas ­ Necessary for species but not individual survival ­ Estrogen pulsatile secretions relates to female sexuality ­ Testosterone is necessary for male copulation ­ Almost all men have the same sexual response curves; women’s vary ­ Brain areas o Male  Mostly VTA (addiction)  Oxytocin release  Sexually dimorphic nucleus (in MPOA = INH3) o Female  Entire brain involvement (some activation/some inhibition)  Activates deep cerebellar nuclei  Inhibits orbitofrontal cortex (decreased judgement)  No VTA activation  Ventromedial hypothalamus (hunger and receptivity to advances) o Both - medial amygdala (dimorphic) ­ NT o Dopamine – excitatory (activates MPOA/INH3) o Serotonin – inhibitory Homeostatic systems ­ Use negative feedback; defend a set point Thirst ­ Hypovolemic o Loss of volume (water and salt)  dec. BP (detected by baroreceptors)  renin from kidneys  inc. in angiotensin II  constriction of vessels (by vasopressin  raised BP), reduced flow to bladder, and thirst (by triggering subfornical organs)  Pituitary adenoma – too much vasopressin  Diabetes – no vasopressin, too much flow to bladder  chronic thirst ­ Osmotic o High salt concentration (osmolarity) sensed by mechanoreceptors  osmosensory neurons in OVLT (hypothalamus) shrink and open sodium channels  release of antidiuretic hormone by the pituitary Hunger ­ Glucose (principal fuel) – stored as glycogen (short-term) or lipids (long-term) ­ Hunger involves multiple signals which are integrated to decide hunger or not o Involves external factors (emotions, food desirability, environment, etc.) ­ Pro-hunger o Ghrelin- made by stomach cells o Cortisol o NPY/AgRP neurons in arcuate nucleus (hypothalamus)  Make neuropeptide Y and agouti-related peptide ­ Anti-hunger o Leptin – made by fat cells; measured by hypothalamus; blocks AgRP o Glucose, insulin, norepinephrine o PMC/CART neurons in arcuate nucleus  Make pro-opiomelanocortin and cocaine- and amphetamine-related transcript (CART) ­ Hypothalamus o Hunger control center o Lateral hypothalamus (LH): pro-eating, helps control set point o Ventromedial hypothalamus (VMH): anti-eating, helps control set point o PVN (periventricular nucleus) integrates signals from LH and VMH o Arcuate nucleus (in hypothalamus) contains 2 sets of opposing neurons (NPY/AgRP vs PMC/CART) Eating disorders ­ Anorexia o Continually lowering set point o Have a larger insula (disgust) and orbitofrontal cortex (tells you when to stop eating) o Commonly occurs with other psychiatric disorders ­ Bulimia o Different than anorexia; recurrent binge eating and compensatory behavior ­ Obesity o Brain disorder – derangement in homeostasis o Occurs with Prader-Willi syndrome (elevated ghrelin) o Strenuous/frequent exercise o Treat as an addiction  Obese people have fewer dopamine receptors and lower prefrontal lobe metabolism o Gastric bypass reduces ghrelin Metabolism ­ BMR (basal metabolic rate) = energy required to fuel brain/body/maintain temp ­ BMR falls starting a diet to prevent weight loss ­ 40% decided by heredity ­ Increased by activity and drugs Emotions ­ Emotion = a change in physiological activity accompanied by related feelings ­ Studied using fMRIs ­ Model - 2 axes (X: avoidance vs approach behavior; Y: Intensity of emotion) o Only 8 main emotions but variations in intensity ­ Stimulus  cognitive appraisal  effects (autonomic and somatic) o Feedback – effects change emotional feelings which change cognitive appraisal ­ Culture o Emotions are pan-cultural (biological)  Culture regulates end result (how you express an emotion based on what is “appropriate” to that culture/situation) ­ Facial expression impairment limits social interaction o Moebius syndrome  facial paralysis/ Bell’s palsy  half of face paralyzed ­ Facial feedback o Facial expressions tell the brain how to feel  Fake smile  real happiness  Sad face – makes pain seem more intense  Can’t frown (botox) – have less amygdala activity = less anger Brain areas – control the intensity of the emotion ­ Locus coeruleus – site of norepinephrine synthesis (moderate pleasure/content) ­ VTA/substantia nigra - dopamine (extreme pleasure/exhilaration) ­ Orbitofrontal cortex (“controller”) (=prefrontal/frontal lobe) o Activated by laughter and “shock value” (cursing) o Inhibits raw emotion from the amygdala o Case: Damage  can’t feel emotions o Last place before action (judges consequences)  May suppress laughter in church, etc. o When damaged can understand social rules but can’t apply them o Abnormalities associated with aggression, depression, and schizophrenia ­ Limbic system (“integrator”)- Integrates info to make decision ­ Hypothalamus - ANS reactions (heart rate, pupils, etc.) ­ Amygdala – fear and aggression o Memory formation and recognition of fear in other people o Amygdala damage  overly trusting people o Anxiety-reducing drugs act on the amygdala ­ Hippocampus- memory (emotional, spatial) o Experience determines the severity of the situation ­ Cingulate gyrus (= anterior cingulate cortex/empathetic nervous system) o Pain processing and empathy o Combines all information in a conscious emotional experience ­ Pathway o Stimulus (sensory organs)  activate thalamus, sensory cortex, and hippocampus  amygdala  hormones and hypothalamus (ANS and emotional behavior) o The amygdala is the link between the hippocampus and emotional behavior (hypothalamus) Aggression ­ Intended to harm; not always physical ­ Can activate reward/addiction areas (amygdala and striatum/nucleus accumbens) ­ Reactive aggression - emotional and unplanned (ex: road rage, murderers) o Lower activity in prefrontal cortex  Less gray matter in prefrontal cortex is associated with antisocial personality disorder o Reckless, overreact, sexually promiscuous ­ Proactive aggression is premeditated, unprovoked, emotionless (ex: planned killings) o Associated with psychopathy (no remorse) o Less autonomic response to stress o Impaired amygdala function ­ Hormones o High testosterone levels associated with aggression/criminality  Aggression and competition (winning) increase testosterone  Testosterone increases dopamine (addictive) ­ Brain areas o Murderers have higher activity of the amygdala and hypothalamus o Tumors in amygdala, hypothalamus or septal area can cause aggression o Seizure activity in the amygdala increases aggression Stress ­ Responses o Activates autonomic nervous system (ANS)   high cortisol (adrenal cortex), low testosterone, and high epinephrine and norepinephrine (adrenal medulla) o Hypothalamus activates sympathetic nervous system (part of ANS)   inc. heart rate, blood flow, breathing ­ Acute stress o Hypothalamus and pituitary stimulate adrenals to release epinephrine and norepinephrine (increases output from the heart and glucose) and cortisol (increases energy) o Beneficial - boosts immune system and brain builds new neurons o Harmful- can lead to heart attacks ­ Chronic stress o Negatively impacts many areas; compromises the immune system (noradrenaline) o PTSD – involves the limbic system (amygdala- fear)  Symptoms include re-experiencing, avoiding triggers, hyperarousal  Low success rate for treatment (20%)  Behavioural desensitization – repeatedly exposing to triggers; stressful  Propranolol – may help block memory of event  Ecstasy – similar to SSRI but much more potent ­ Damages (reduced volume) the… o Frontal cortex (executive function)  Can lead to inappropriate behavior (abuse, cycle of abuse, etc) o Hippocampus, cortical tissue, amygdala (enlarges then atrophies) o Damage due to stress is caused by cortisol Other emotions ­ Pleasure/happiness - Related to craving and addiction o Involves medial forebrain bundle – contacts midbrain, hypothalamus, locus coeruleus, VTA, and nucleus accumbens ­ Surprises - involves the nucleus accumbens (NAC) – addiction o Unexpected surprises  more activation ­ Klüver–Bucy syndrome - lesions in temporal lobe o Removal of temporal lobes  socially unacceptable behavior  Failure to recognize other’s emotions o Studies showed that the amygdala is important in fear ­ Disgust and guilt activate the insula ­ Romantic love o New love activates nucleus accumbens and VTA (“lust”) o Old love – lust areas do not activate o Septal nucleus (limbic system) - pleasure, sexual fantasies, and arousal Physiological basis of personality ­ Introverts (social anxiety) have higher norepinephrine (NE) - blocks immune system ­ Person with higher activity in the L hemisphere had a better response to the flu shot Two hemispheres ­ Left side of the face (R hemisphere) is more expressive  R brain - more emotional ­ Right hemisphere identifies tone of voice ­ Left hemisphere processes words/meaning of vocal message ­ Left frontal lobe o Activated  behavioral approach  More active during positive emotions  Damage to the left hemisphere  anxiety and sadness o Regulates anger and joy ­ Right frontal lobe o Activation  behavioral withdrawal  More active during negative emotions o Regulates fear and sadness o Damaged  patients more likely to be unbothered or euphoric Sensory processing ­ Stimulus to receptor produces a graded potential (EPSP) o If big enough  action potential o Not big enough  sensory receptor still feels it even though the brain didn’t Processing intensity ­ Both light touch and pain signal an action potential (all the same) ­ 3 strategies; brain can know intensity based on… o 1) How fast it fires  A single neuron increases the frequency with increased intensity o 2) How many fire  Multiple neurons fire together (more neurons with inc. intensity) o 3) Which neurons fire  Different neurons respond to different ranges  Range fractionation - “specialized neurons”  Different neurons fire based on intensity; brain knows how intense based on which neurons fire  Neurons limited in intensity because of refractory periods Pathways of processing ­ Any sensory input goes to the thalamus  primary sensory cortex  non-primary (on both sides) to process signals ­ 2 systems o Dorsal for touch and vibration (do NOT use free nerve endings; use 4 other) o Spinothalamic for pain and temp (use free nerve endings)  Skin  spinal cord PAG  thalamus  cingulate cortex and somatosensory cortex  PAG (periaqueductal gray) – midbrain; has opioid receptors (endorphins); can block pain transmission if necessary  Cingulate cortex/gyrus – Branch off pathway from skin to somatosensory cortex  Part of limbic system - emotional memory (of pain)  Opioids take away the emotional quality of pain not the actual pain  Increasing pain: spinal cord  thalamus  Decreasing pain: limbic and hypothalamus  PAG Somatosensory system ­ Detects touch and pain (different/independent systems) ­ Dorsal root ganglion –involved in all sensation o Cell bodies of the sensory neurons throughout the human body ­ Somatosensory cortex – “the homunculus” o Analyzes all sensory info in the postcentral sulcus/gyrus o Postcentral sulcus – divided into regions – one for each part of body (cortical map); smaller receptive fields = bigger area in cortex o After extremity is amputated, the other areas take over ­ 4 receptor classes- pain and touch (epidermis); vibration and stretch (deeper) ­ 5 types of receptors o 1) Free nerve endings – sample extracellular fluid (pain and temp) o 2/3) Merkel’s disc and Meissner’s corpuscle – very precise o 4/5) Pacinian corpuscle (vibration) and Ruffini’s ending (stretch)- less precise ­ Temperature (pain pathway) o Temperature receptors (free nerve endings)  Cold receptor – CMR1, warm receptor – TRPV1, hot receptor – TRP2 o Pain pathway uses 2 types of neurons  Small unmyelinated C fibers- dull pain; most common; cold/warm  Dull because not myelinated (conducts slowly)  Signal from spinal cord through dorsal to thalamus  Capsaicin (chili pepper) activates via TRVP1 and kills fibers  Larger myelinated A-delta fibers- sharp pain; hot  Injury is a “2 phase” response: A-delta fibers then C fibers ­ Vibration o Oldest/simplest receptors; unipolar o Vibration stretches skin – opens sodium channels  system fires ­ Touch – 2 parallel systems (early or later touch) st o 1 phasic then tonic o Phasic receptors – Burst of intense/fast firing to grab the brain’s attention  Adapt quickly by decreased frequency o Tonic receptors – Fire from onset, keep firing after phasic stop  Slow or no decline in action potential frequency ­ Pain o Involves sensation and other effects; multisystem reaction o Free nerve endings (pain receptors) fire action potential to spinal cord which transmits info to brain where pain processing occurs o Cognitive system – involves frontal cortex o Motivation-affective – leads to motor response  Can block pain signals o Sensory-discriminative – brain identifies characteristics (change in pain/which side hurts) o Peripheral mediation (other effects)  Trauma  chemical release  Histamine swelling  Substance P- transmits pain  Serotonin- signals C fibers to fire o Disorders  Congenital insensitivity – mutated Na+ channel in pain fibers  Congenital hypersensitivity (“man on fire”); gain of function mutation in Na+ channel o Mild pain prevents bedsores & losing feeling in an area from lack of blood flow o Gate control theory  Blocking pain by deep pressure – activates A-beta fibers which closes the gate from c fibers to spinal cord  TENS – electrical nerve stimulation; activates A fibers o Chronic pain – pain after injury has healed because pain signals remain active  Has emotional effects  2 kinds: neurogenic (nerve damage) and psychogenic (no damage) o Neurogenic pain = reflex sympathetic dystrophy  Stage 1 (normal) –pain from injury  Stage 2 (abnormal/chronic) – pain; swelling; muscle wasting  Mechanism  Injury  released chemicals to CNS to trigger the sympathetic nervous system which releases the same chemicals back to injured area for inflammatory response and more pain o Cycle:  more signaling to SNS  more signaling back  Occurs over months  More likely to happen in young females (higher activation of sympathetic nervous system)  Treatments  Antidepressants, cut sympathetic nerve to injured area  Anti-inflammatory to block chemical release o Chronic pain remodels the spinal cord and brain  The spinal cord learns to be in pain (= inappropriate pain signaling)  This causes neurogenic pain  Spinal cord neurons take up substance P (released during pain) and remodel their dendrites  Non-pain and painful dendrites become cross-wired – what stimulates touch (non-pain) receptors also stimulate the pain receptors  Dorsal horn neurons (light touch) become hyper-excitable  Somatosensory cortex is rewired in chronic pain (over- active)  What makes it go away  Spinal cord/brain remodel themselves after a while  Pain becomes an emotion in the cingulate cortex (activates when somatosensory cortex reaches max activity) Receptive field ­ The part of the world a sensory neuron responds to ­ Smaller in areas that are more precise (hands/tongue) ­ Touch receptive fields “on center, off surround” o Touch in center of field = max firing; excitatory o Touch in surrounding area = dec. firing; inhibitory o Touch far away = no effect (out of receptive field) Sensory integration disorder ­ Most common in children; continuum- all people have it ­ 1) Hypersensitivity to touch (stimulus avoiders) o Light touch activates the pain pathway o SID is part of autism but not enough for a diagnosis of autism ­ 2) Hyposensitivity to touch (stimulus seekers) o Less common o Ranges from tapping foot to banging head and biting/hurting kids o Correlated with hyperactivity Synesthesia- stimulus creates sensation in wrong modality ­ Cross linkage between sensory systems (any 2, not all) ­ Causes: LSD, posterior temporal lobe seizure, blindness/deafness, spontaneous Basics of sound ­ Sound is vibrating air that the brain can detect ­ Amplitude = Intensity/loudness; measured in decibels (dB) ­ Frequency = pitch; cycles per second; measured in hertz (Hz) o Cats hear higher frequencies o Elephants hear low frequencies because of large ears – hear footsteps o Mothers hear higher frequencies than other people o Humans hear about 20 Hz – 20K Hz  Conversation range is 2000Hz or less  Speech evolved to be in the best range for hearing ­ Pure tone - single frequency of vibration; electronic; not interesting to brain ­ Musical tone – modulated pure tones with repetition (rhythm); involves a pattern ­ Noise – random sounds; un-patterned; most sounds ­ Fourier analysis o Complex sound = sine waves added together o Brain breaks sounds down to individual waves o More waves (hundreds) = probably noise Parts of the ear ­ Outer ear: The external ear and ear canal - collect low frequency sound waves ­ Middle ear: Up to tympanic membrane (= ear drum) ­ Inner ear: cochlea (including oval window which is in cochlea “membrane”) ­ Two muscles in the middle ear protect hearing o When activated too much, they stiffen/retract so that they don’t vibrate the oval window even when sound is coming in to the eardrums o Protection because loud sounds kill hair cells which cannot be repaired ­ Auditory hair cells o Inner – “transducers”; used in understanding speech  Die over time with age o Outer- “amplifiers”; hearing support (not as important)  Adjust the volume by increasing the volume of the signal and noise  Makes it difficult to distinguish sounds o Neurons run to the ear from the brain and back (feedback) o Hair cells do not have axons - no action potentials (graded only) Hearing pathway ­ Air molecules vibrate and come through the ear canal which vibrates the ear drum ­ Ear drum vibrates bones which causes the oval window to vibrate ­ Causes basilar membrane in the cochlea to vibrate which causes the stereocilia on hair cells to bend o Hair cells are stuck in the tectorial membrane so that when the basilar membrane moves, they bend ­ Stereocilia bending causes ion channels to be mechanically pulled open by tip links that connect all the stereocilia on a hair cell to one another ­ Causes ions to come in = depolarization and calcium influx  glutamate release ­ From ear to brain: o Auditory nerve  cochlea  cochlear nucleus  superior olivary (SO) nuclei  inferior colliculus  medial geniculate nucleus (in the thalamus)  auditory cortex  SO nuclei is the first place with bilateral input  Most right ear sounds are processed by the left hemisphere but the process is bilateral Auditory cortex ­ Processes sounds- distinguish noise from patterned sound then Fourier analysis ­ Auditory cortex cells each have a ‘preferred frequency’ (receptive field) o Some cells are less picky than others – larger receptive field  Trying to pick up any sound  Good at detecting whether sounds are changing ­ Auditory neurons have tonotopic organization (arranged by tone)  Cells that detect high frequency are on the inside, in the back  Low frequency sounds are processed by the anterior auditory cortex  Neurons next to each other are similar frequencies ­ Auditory cell’s receptive field can shift if needed (plasticity) ­ Auditory cortex analyzes complex sounds in two streams o Dorsal stream (frontoparietal lobe) – used in sound location o Ventral stream (temporal lobe)- analyzes “what is it” ­ Speech is analyzed much more than random noise ­ Left hemisphere processes sounds that you know; R processes sounds you don’t Two ways that we hear pitch (frequency) ­ 1) Frequency coding - encodes pitch by the firing rate o Only works for lower pitches because there’s a max rate o Volley principle  Each fiber only fires at a certain point in the cycle but does not respond to each cycle – all together = “complete transcript” ­ 2) Place coding - Brain knows frequency by which part of the basilar membrane vibrated o Not as accurate as frequency coding o The cochlear apex is shaped like a triangle; skinny end = high pitch  Different thickness of string = different frequency ­ Place and volley mechanisms act together ­ 1. Low frequencies - frequency coding ­ 2. High frequencies - place coding ­ 3. Intermediate frequencies (1000-5000 Hz) use both Detecting sound location ­ Binaural cues detect sound location ­ Compare the ears in intensity (which is loudest) and latency (which got here first) o Mainly uses superior olive (SO) ­ High frequency - intensity differences in the lateral SO (lateral compares loudness) o Travel too fast to tell difference in arrival times ­ Low frequency - latency differences in the medial SO (medial compares meeting times) o Notes where the input from the left ear and right ear meet in the MSO o If the sound is from the far right side, right ear is farther along in processing so sounds meet closer to left Unconscious hearing ­ Involves three systems: sleep, limbic system, and hypothalamus ­ Separate pathway than conscious hearing Hearing loss ­ Central deafness - rare; hardest to treat o Caused by problems (lesions, injuries) in the brain o Associated with neurologic disorders (multiple sclerosis, tumors) o Involves CNS (cortex, brainstem, or ascending auditory pathways) o 1) Cortical deafness  Pure word deafness  Can speak normally and identify nonverbal sounds  Can’t understand speech  Auditory agnosia  Can only hear pure tones  Cannot recognize verbal or nonverbal sounds o 2) Auditory hallucinations  Hearing things that aren’t there  Common in schizophrenia, injury to secondary auditory cortex, or during a temporal lobe seizure  More common than visual hallucinations in schizophrenia  Usually negative towards themselves (danger or insults)  Can be caused by damage to brainstem structures (like SO) ­ Conduction deafness o Disorders of outer or middle ear - sounds don’t reach cochlea o Causes- otitis Media (swelling of the eardrum), TM Perforation, or ossicular arthritis (small bones can’t move/conduct properly) ­ Sensorineural deafness = SNHL (sensorineural hearing loss) o Problem with cochlea, auditory nerve, or hair cells o May cause loud voice (can’t hear themselves) o High frequency loss common  speech sounds distorted o Can be caused by viral infections (measles/CMV) that kill developing hair cells Other hearing problems ­ NIHL (noise induced hearing loss) - #1 preventable cause of deafness o Can be sudden or gradual (accumulated damage over time) ­ Presbycusis o Gradual onset hearing loss; starts at age 30 o As age increases, higher frequencies have to be louder ­ Tinnitus o Ranges from moderate to severe (suicidal) o Damage to cochlea initiates tinnitus, CNS maintains chronic tinnitus o Brain problem, not a hearing problem (maladaptive memory)  = an auditory hallucination All languages have similar basic elements ­ Phonemes – basic speech sounds ­ Semantics – meanings of words or sentences ­ Syntax – grammatical rules for constructing phrases and sentences Learned or innate ­ From birth, babies can distinguish phonemes in any language ­ Language dev. depends on experience during a sensitive period (first few years) Basic disorders of language ­ Dysphasia/aphasia– any language disorder; impaired language ability ­ Dysarthria – inability to speak clearly (muscle control problem) o “Speaking with marbles in your mouth” ­ Dyspraxia – inability to sequence a complex motor act o Intrusion error – trying to say one phoneme and another comes out ­ Dysgraphia – inability to write ­ Dyslexia – inability to read Types of aphasia ­ Paraphasia – substitution for a word by an incorrect, unintended word o "The Lord is a shoving leopard“ ­ Neologism - an entirely novel word (person can’t recall the correct word) ­ Broca’s (non-fluent) aphasia o Most common type of aphasia ­ Can understand other people’s speech ­ May occur with anomia ­ Continuum; ranges from moderate to severe ­ Affected area = Broca’s area (right next to motor cortex) ­ CQ: A stroke in Broca’s area causes… ­ Non-fluent aphasia ­ Ex: Severe case ­ “Tono” aphasia ­ Can say the numbers 1-10 but everything else is “tono” ­ Intonation is normal ­ Caused by a left hemisphere stroke – can’t move right hand ­ Can understand what someone is saying but can’t understand his own speech ­ Ex: Less severe case ­ Patient can hear himself, knows that his speech is messed up and is trying to fix it but can’t ­ Ex: CQ ­ Woman is asked where her son is ­ She answers “Car..home..yes” in response ­ She has a less severe case of Broca’s aphasia ­ Wernicke’s (fluent) aphasia ­ Complex verbal output with many paraphasias ­ AKA – “word salad” ­ Ex: “Train flogging into my question, are you why is it thirty letters down under peanut butter, what is it” ­ Patients cannot understand what they read or hear ­ Left brain problem - affects Wernicke’s area (temporal area) ­ Can tell that someone asked a question because they can hear the intonation but don’t understand the question ­ Global aphasia = Broca’s + Wernicke’s ­ Total inability to understand or produce language ­ Large left-hemisphere lesions, affecting all speech zones ­ Prognosis is poor ­ Caused by horrible trauma ­ Conduction aphasia ­ Impaired repetition of words only ­ Lesions of arcuate fasciculus axons produce conduction aphasia ­ Arcuate fasciculus – axons connecting Wernicke’s area to Broca’s area ­ Supports Wernicke-Geschwind model that damage to connections (axons) causes language impairment ­ Anomia ­ Cannot name a seen object or speak a written word ­ Naming a seen object requires transfer of visual information to the angular gyrus ­ Lesions disconnect visual and auditory systems ­ Patients can speak and understand, but can’t read aloud Lectures 15 and 16: Language and Cognition All languages have similar basic elements ­ Phonemes – basic speech sounds ­ Semantics – meanings of words or sentences ­ Syntax – grammatical rules for constructing phrases and sentences Learned or innate ­ From birth, babies can distinguish phonemes in any language ­ Language development depends on experience during a sensitive period (first few years) o Japanese babies can understand the distinction between R and L but raised in Japan with no exposure to English they lose this ability Basic disorders of language ­ Dysphasia /aphasia– any language disorder; impaired language ability ­ Dysarthria – inability to speak clearly o Muscle control problem – “speaking with marbles in your mouth” ­ Dyspraxia – inability to sequence a complex motor act o Intrusion error – trying to say one phoneme and another comes out ­ Dysgraphia – inability to write ­ Dyslexia – inability to read Types of aphasia ­ Paraphasia – substitution for a word by an incorrect, unintended word o "The Lord is a shoving leopard“/“The light at end of the candle” o Common to occur every now and then but in the case of paraphasia it’s functionally impairing ­ Neologism ­ an entirely novel word o Affected person has to make up words because they can’t recall the correct word Broca’s (non­fluent) aphasia o Most common type of aphasia o Can understand other people’s speech o May occur with anomia o Continuum; ranges from moderate to severe o Affected area = Broca’s area (right next to motor cortex) o CQ: A stroke in Broca’s area causes…  Non­fluent aphasia o Ex: Severe case  “Tono” aphasia  Can say the numbers 1­10 but everything else is “tono”  Intonation is normal  Caused by a left hemisphere stroke – can’t move right hand  Can understand what someone is saying but can’t understand his own speech o Ex: Less severe case  Patient can hear himself, knows that his speech is messed up and is trying to fix it but can’t o Ex: CQ  Woman is asked where her son is  She answers “Car..home..yes” in response  She has a less severe case of Broca’s aphasia Wernicke’s (fluent) aphasia o Complex verbal output with many paraphasias  AKA – “word salad”  Ex: “Train flogging into my question, are you why is it thirty letters down under peanut butter, what is it” o Patients cannot understand what they read or hear o Left brain problem ­ affects Wernicke’s area (temporal area) o Can tell that someone asked a question because they can hear the intonation but don’t understand the question ­ Global aphasia = Broca’s + Wernicke’s o Total inability to understand or produce language o Large left­hemisphere lesions, affecting all speech zones o Prognosis is poor o Caused by horrible trauma ­ Conduction aphasia o Impaired repetition of words only o Lesions of arcuate fasciculus axons produce conduction aphasia  Arcuate fasciculus – axons connecting Wernicke’s area to Broca’s area o Supports Wernicke­Geschwind model that damage to connections (axons) causes language impairment Anomia o Cannot name a seen object or speak a written word o Naming a seen object requires transfer of visual information to the angular gyrus o Lesions disconnect visual and auditory systems o Patients can speak and understand, but can’t read aloud Anatomy of Aphasia • Wernicke­Geschwind model = disconnection theory • Language impairment from loss of connections among brain regions involved in language • Explains most clinical aphasias Recovery from aphasia ­ If injury to left hemisphere occurs early in life, the right hemisphere can take over language o Can have completely normal recovery ­ If damage occurs later, language control is more likely to shift into bordering areas in the left hemisphere o Right hemisphere won’t take over; the brain has to rely on what is remaining in the left hemisphere but can still get some recovery o Plasticity is severely diminished later in life ­ Ability of other areas to take over language functions (either right or bordering areas depending on age of injury) may be due to their normal participation in language o Indicates that the right hemisphere may be more involved in language than previously thought ­ Melodic intonation therapy (MIT) for aphasia uses the fact that singing is often intact after a left hemisphere stroke o Intact because music is processed by the right hemisphere o In therapy, they try to use music to incorporate language ASL ­ Uses the same brain mechanisms as spoken language o Lesions that cause aphasia in normal speakers also cause impairment of ASL use in the deaf ­ fMRI studies show that signers and normal speakers have similar activation in language tasks but signers have extra activation in the right hemisphere Mirror neurons ­ May be critical to develop language ­ Active during imitation of others’ actions, not experience of themselves speaking o Babies learn speech by watching others speak ­ Overlap Broca and Wernicke areas Brain looks at whole words – not every letter ­ As long as the first and last letter of a word are the same, you can read a sentence where every middle letter is rearranged Bilingualism ­ If a second language is learned before age 11, the same brain regions (angular gyrus) are used for both languages o Angular gyrus is the “dictionary” of the brain ­ If learned after age 11, different regions are used o Angular gyrus used for the first language and the insula for the second ­ Language acquisition is more difficult in adulthood, reflecting diminished plasticity ­ Easier to learn a language earlier in life but may be beneficial to learn it later – if a stroke wipes out the angular gyrus you can still speak Identifying words via tactile ASL ­ Can use braille; but they’re feeling someone’s hands as they’re signing ­ Regions involved in vision become used for tactile stimulation in a blind person The brain activates more with real speech ­ More active with real word than real sounding word than random letters than characters Brains use two language systems to read ­ Learned because of dyslexia ­ One focuses on sounds of letters, the other on meanings of words Dyslexia ­ Visual AND auditory processing disorder ­ Surface dyslexia ­ errors in reading restricted to details and sounds of letters o Pretty read as “PRI­tee” ­ Deep dyslexia ­ errors in reading one word as another o The words are related in meaning o They substitute a word for a synonym that’s easier to say o Caused by extensive damage to left­hemisphere language areas ­ Dyslexics have distortions among cells in the cerebral cortex o In the planum temporale there is micropolygyria (excessive cortical folding) and ectopias (clusters of extra cells) o CQ: some dyslexic brains have…  Too many neurons along the surface of the brain ­ Dyslexic brain uses more of the right hemisphere than a normal brain to read ­ Dyslexia improves with training and brain changes occur o Before remedy, there is little activation in Wernicke’s and Broca’s area o After remedy, there is a lot of activation in both Cerebral lateralization* ­ Left face is more expressive because emotional expression is in the right hemisphere ­ Arithmetic – left cortex even though it’s not language (but it’s a language of sorts) ­ Geometry (not a language) is in the right hemisphere ­ Fourier transformations (for dissecting sounds) are in the left hemisphere Corpus callosotomy ­ Cutting corpus callosum (connection between the hemispheres) to prevent seizures from spreading ­ Allows study of each hemisphere independently (=split­brain patients) Hemispheres of split­brain patients function independently ­ Left hemisphere can tell what it sees ­ Right hemisphere can show what it sees ­ In split­brain patients: o Present a picture to the right visual field (stimulate left brain)  Left hemisphere can tell you what it was  Right hand (left hemisphere) can show you, left hand can’t o Present a picture to the left visual field (stimulate right brain)  Subject will report they do not know what it was  Right brain can’t tell you what it is  Left hand (right hemisphere) can show you what it was, right hand can’t  Right brain can only show you Another way to test the hemispheres independently ­ Dichotic listening – presents different sounds to each ear at a different or same time ­ Sound into left ear has to go across the left to the right hemisphere then back to the left ­ Sound into right ear goes across right straight to left hemisphere o = More efficient; right ear advantage ­ Shows that the left hemisphere is dominant for language ­ If you play both at same time – you’ll only hear the right ear because it’s more efficient and thus overtakes the left ear ­ Right­handers identify verbal stimuli delivered to the right ear more easily ­ Some left­handers have left­ear advantage o This implies some left­handers have language in their right hemisphere Wada test ­ Injection of amobarbital into the carotid artery briefly sedates that hemisphere ­ Right­handers: Language restricted to the left hemisphere in 95% of patients ­ Left­handers: only 70% have left lateralization, 15% have right lateralization (right dominant) and 15% mixed dominance Structural asymmetry of the planum temporale ­ Generally larger on the left ­ Asymmetry is present by week 30 in gestation ­ Support for innate language o Innate verbal specialization of the left hemisphere for language ­ Chimpanzees also have this but people can’t get them to talk so maybe support against innate hypothesis Music and the right hemisphere ­ Auditory areas of the right hemisphere play a major role in perception of music ­ Musical perception is impaired by damage in the right hemisphere ­ Music activates the right hemisphere more than the left (some activation on left) ­ However, simple tones and perfect pitch involve the left hemisphere Cognition ­ Human cortex is divided into 4 lobes: frontal, parietal, temporal, and occipital lobe o Lobes are named for bones not for function; it just so happens that the division of lobes based on the bones also matches with division of function ­ The little man inside the guy’s head is called a homunculus – meaning “little man” Two functional types of cortex ­ Primary cortex (pink): First cortical area processing sensory input or motor output o Processes the simple question: What is it o Evolved first so it represents all of the primary cortices (primary visual, auditory etc.) o Motor cortex – once association cortex decides what to do, it carries out the function ­ Association cortex (blue): Multi­modal cortex (most of cortex­ covers more area) o Association areas – visual association area, etc. o Processes complex question: what does it mean o Acts as commander: What do I do now ­ Info starts in primary (what is it) then travels to association (what does it mean/what do I do) and then back to primary (do it­ involves motor cortex) ­ The areas are not 100% accurate which can lead to hallucinations or illusions ­ The brain changes information based on genetics, experiences, etc. Frontal Lobes: Anatomy ­ Everything in front of the central sulcus ­ 1) Motor cortex (blue) ­ 2) Premotor cortex (green) ­ 3) Prefrontal cortex (purple) o Makes decisions o Divided into 2 partitions  Dorsolateral – logical/reasoning­type thinking  Not connected to amygdala so no role in emotion  Inferior prefrontal cortex = orbitofrontal  Located really close to the amygdala = more emotional Frontal lobe damage ­ Phineas Gage ­ railroad worker ­ Explosion drove a tamping rod up through left cheek and out the top of his head o Serious left eye damage but most people injured by a tamping rod don’t survive ­ Damaged orbitofrontal cortex bilaterally ( trouble regulating emotions) ­ Prior to accident, Gage was a sober, serious workman; afterward, he was moody and uninhibited ­ Abuse/stress  cortisol  frontal cortex damage w/o trauma Symptoms of frontal lesions ­ Children (5 year olds etc.) whose frontal lobes have not yet developed also have frontal lobe issues (similar to trauma patients) ­ Perseverative errors ­ Failures of planning o Like a 5 year old; impulsive; like ADD ­ Reduced pain reactivity o Prefrontal cortex connects pain perception and sensation o Children can break their arm, cry and then be fine 10 seconds later because they have reduced pair sensation ­ Inappropriate social behavior o A child asks a stranger why they’re fat ­ Attention and memory problems, though little change in IQ score o Prefrontal cortex is not responsible for “smart” ­ Dorsolateral damage ­ poor judgment and self­care, motor programming problems o Damage doesn’t affect emotion o Involved in guilt o People drink alcohol affects inner monologue because it directly affects the frontal lobe which leads to less anxiety ­ Orbitofrontal damage ­ environmental dependence, poor social insight, emotional damage ­ All cortexes talk to the frontal cortex Parietal lobe organizes space ­ Parietal lobe injuries produce diverse impairments – it touches all three other major lobes ­ Touches all other lobes, so it’s logical that it integrates/is involved in sensory integration o Ex: body’s position in space, body image, phantom limb ­ The least is known about the parietal lobe ­ Injury to postcentral gyrus (somatosensory cortex) results in tactile deficits on the opposite side – but not anesthesia ­ Astereognosis – inability to recognize objects by touching and feeling them – mostly right hemisphere o People could tell what an object is by feeling it without looking at it because they create a map of the object and put it together in their head Hemispatial neglect ­ Damage to right parietal lobe results in neglect of the left side of the body and space ­ Diagnosis – patient tries to copy picture and only draws the right side o If you ask them where the left side of the picture is, they’ll say “it’s right there” because they know it’s supposed to be there o It doesn’t make sense to them why it’s not there in their drawing when their experience tells them it should be there Prosopagnosia ­ People fail to recognize familiar faces, even theirs ­ Recognizing faces is a learned behavior o Involves the visual parietal cortex ­ Ex: Man couldn’t tell if his wife was a wife or a hat until she started talking because they have roughly the same shape ­ Bilateral damage to fusiform gyrus causes prosopagnosia o Fusiform gyri is specific for human faces ­ Often accompanied by other forms of agnosia (inability to identify items) Case ­ Man felt ‘someone’s leg’ in the bed ­ Tried to throw it out of bed but he came after it and it was now attached to him ­ Tried to pull it off ­ Someone told him it’s his own leg and he didn’t believe it ­ They asked “if it’s not yours then where is your left leg” ­ He said it had disappeared ­ Always happens to the left leg because it’s caused by damage to the right parietal lobe o Left parietal stroke/damage – not much change o Right parietal lobe is involved in emotion, body map, and spatial reconstruction Lecture: Attention ­ Why don’t we always know where our tongue is in our mouth We don’t notice until asked ­ Your brain’s job is to constantly scan the environment to answer 3 questions: o Will it kill me Can I eat it Can I have sex with it Attention is how we filter out unimportant details ­ All the incoming stimuli is more than you can process so the brain has to choose what to pay attention to ­ A decision has to be made very fast – brain immediately goes through the 3 questions ­ 2 systems decide what to pay attention to Top­down (left side on picture) ­ Dorsolateral frontal cortex = dorsal frontoparietal system = dorsal stream o Connected by axons to parietal o Parietal – special awareness/frontal – makes decisions ­ AKA voluntary attention ­ The dorsal network allocates attention under goal­directed control ­ A decision is made (rules set) before you experience the stimuli – “I’m going to pay attention in class today because the test is next week” o This screens out all the other information and you only pay attention to what will be useful for the test o Uses prior knowledge – have to know what will be useful ­ Higher­order cognitive processes exclude unattended information before any perceptual processing has occurred o Higher analysis controls what you take in ­ Problem – what if you pick the wrong rule o Ex: You listen for “test” and miss parts when he says “exam ­ Example: not hearing your girlfriend when the Cowboys are first­and­goal o Dorsal says pay attention to auditory and visual channel and ignore touch stimulus so you don’t notice when your girlfriend is tapping you on the shoulder o Problem – maybe your brain made the wrong decision in putting the game above your girlfriend Bottom­up (right side on picture) ­ Temporoparietal system = ventral stream ­ AKA involuntary attention ­ No decision made ­ Process everything at a shallow level and then decide what to pay more attention to based on change ­ This system is activated by novel or unexpected stimuli ­ The more ventral network responds to stimulus demands ­ Useful in situations where you don’t know the “rules” o Ex: first day of college – you don’t know what’s important for the test ­ You try to take in everything but you can’t ­ a decision still has to be made o So, the brain looks for novelty (change) ­ Ex: if you have chronic pain, feeling pain is not helpful but if it’s a new pain it’s worth paying attention to ­ Ex: paying attention when your girlfriend pours your beer on your head o That’s unexpected enough that you notice o But your girlfriend taps on your shoulder all the time ­ Ex: child throwing a tantrum over a new bike they want – trying to talk/reason isn’t helpful because you talk all the time – instead “hey look santa claus” is enough to distract them – or picking them up by the heels and carrying them upside down – when you put them down they’ve forgotten all about the bike ­ Ex: commercials constantly change the image ever 150 milliseconds to keep people’s attention Our brain guesses what it sees ­ we see what we expect to see ­ Why have 2 systems ­ Top­down is more efficient/faster o Tries to make it even faster by guessing what it sees ­ What you expect to see changes how you process the stimulation o Biases effect how you see, smell, feel things ­ Top­down processing is a “shortcut” in which early visual inputs are sent, partially analyzed, from early visual cortex to prefrontal cortex (PFC) ­ An interpretation of the crude visual input is generated in PFC and then sent to inferotemporal cortex (IT), subsequently influencing the slower, bottom­up process ­ This shortcut minimizes the number of object representations required for matching, thereby speeding object recognition, but is error­prone ­ People with PFC injury have slower response times, due to only bottom­up processing ­ Ex: Scorpion in boot one day. Next day there’s a brown leaf in the boot. They freak out thinking it’s a scorpion because the brain filled in the blanks with a scorpion o The brain’s primary job is to protect you not represent reality 100% accurately ADD/ADHD: major symptoms ­ Common ­ Everyone has it, but some have it worse than others ­ Situation­dependent ­ The problem isn’t that you aren’t paying attention but that you’re not paying attention to what you want to; you’re paying attention to too much ­ Bottom­up system is dominant and top­down is weak ­ Medications for ADD help all 3, but some medications target only one ­ Inattention ­ easily distracted and doesn't listen ­ fails to finish – doesn't stick with an activity o More important than impulsivity in class situations ­ Impulsivity ­ doesn't wait for turn ­ acts without thinking – can’t organize work ­ speaks out loudly in class o More important in social situations ­ Hyperactivity ­ can't sit still ­ fidgets ­ bothers classmates o Does not imply inattention o Hyperactivity is a way of increasing focus o Because it’s a pattern/predictable, it blocks out other stimuli Consciousness ­ Attention is part of (a defining feature of) consciousness ­ Consciousness involves awareness, attention, and a sense of self (something that distinguishes you from the world) ­ To check for consciousness – you tap (stimulus) and look for a response o If they respond to stimulus they must be conscious Altered states of awareness ­ There are more states than just wake or sleep ­ Arousal – the intensity of response to a stimulus ­ Brain dead – no awareness and no arousal (at intersection of bottom left) ­ Coma ­ not deep sleep; “hibernation state”; o Unarousable and unresponsive to stimulation o By definition, is a short term process (1­2 week process) o After short term – you either die or transition to a vegetative state ­ Vegetative state o Almost no awareness  May respond to a stimulus by a change in heart rate (reflexive response)  No voluntary interaction  In some cases they’re more responsive than we think o Still has arousal – a sleep/wake cycle o Can transition to minimally conscious and in some cases back to normal  Ex: 15% recover from drug exposure that causes a vegetative state ­ Minimally conscious o Some voluntary response but can’t communicate reliably ­ Locked­in syndrome o Full consciousness, but mostly uncommunicative because of paralysis (brainstem lesions) o Can’t move anything but their eyes (because different part of brainstem) ­ Sleep o With increased awareness and arousal – deep sleep then light then drowsiness then conscious wakefulness ­ REM sleep (dream sleep) o Same (low) arousal as deep sleep but more aware ­ Lucid dreaming o Same (low) arousal as deep sleep and REM sleep but increased awareness ­ More states: hypnosis, trance, dreams, meditation, intoxicatio Where is consciousness ­ Consciousness involves a widely distributed network o The whole brain is involved ­ Prefrontal cortex o Active when you become aware of a relationship between objects o Acts as a commander – it makes a decision o Organizes things but not the site of consciousness  If it’s cut out people are still conscious ­ Parietal lobes o Its ability to locate objects in space is needed to combine an object’s

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Chapter 3, Problem 4 is Solved
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Textbook: Vector Calculus
Edition: 6
Author: Jerrold E. Marsden; Anthony Tromba
ISBN: 9781429215084

Since the solution to 4 from 3 chapter was answered, more than 283 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Vector Calculus, edition: 6. This full solution covers the following key subjects: . This expansive textbook survival guide covers 8 chapters, and 305 solutions. The full step-by-step solution to problem: 4 from chapter: 3 was answered by , our top Calculus solution expert on 09/09/17, 04:03AM. Vector Calculus was written by and is associated to the ISBN: 9781429215084. The answer to “Find the absolute minimum and maximum values of the function f (x, y) = x2 + 3xy + y2 + 5 on the unit disc D = {(x, y) | x2 + y2 1}.” is broken down into a number of easy to follow steps, and 34 words.

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Find the absolute minimum and maximum values of the