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(a) A projectile is fired from the origin down an inclined

Calculus | 8th Edition | ISBN: 9781285740621 | Authors: James Stewart ISBN: 9781285740621 127

Solution for problem 4 Chapter 13

Calculus | 8th Edition

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Calculus | 8th Edition | ISBN: 9781285740621 | Authors: James Stewart

Calculus | 8th Edition

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Problem 4

(a) A projectile is fired from the origin down an inclined plane that makes an angle with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are and v0, respectively. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time t. (Ignore air resistance.) (b) Show that the angle of elevation that will maximize the downhill range is the angle halfway between the plane and the vertical. (c) Suppose the projectile is fired up an inclined plane whose angle of inclination is . Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical. (d) In a paper presented in 1686, Edmond Halley summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance R up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in part (c). (Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.)

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Saturday, February 27, 2016 Week 7 Shared Syntactic Integration Resource Hypothesis • language processing in the brain • major structures • primary auditory cortex Broca’s area: speech production • • opercular and triangular parts of inferior frontal gyrus • Wernicke’s area: speech comprehension • posterior portion of superior temporal gyrus • arcuate fasciculus: fibers connecting Broca’s and Wernicke’s area • music processing many areas • • pitch analysis in right temporal • primary and secondary auditory cortices in temporal relations • motor cortical areas involved in rhythm • connection between right temporal gyrus and frontal cortical areas important for working memory in music appreciation inferior frontal gyrus (Broca’s) involved in analysis of musical syntax • • syntactic vs. semantic processing in language • syntactic: how brain combines words into constituents and sentences • Broca’s area (regions in/around inferior frontal gyrus) • superior temporal gyrus/sulcus • left anterior temporal lobe semantic: how structural and semantic information is used in the understanding of • sentences • Wernicke’s area • inferior prefrontal cortex • syntax in music and language: • hierarchically organized sequences of basic elements language: phenomes, morphemes, words • • music: notes and chords • lower-level units arranged according to rules • language: complex words, phrases, sentences • music: motifs, phrases, movements • dependent on long-term memorized representations language: knowing lexical knowledge and the meaning of specific combinations of • words • music: memorizing familiar tunes as specific sequence • dissociation evidence: • aphasia: speech/language impaired in either auditory comprehension, verbal expression, reading/writing or functional communication patients that were aphasic after removal of left hemisphere or after stroke: • • restricted speech • able to recover word articulation with support of melody • due to the fact that language lateralized to left hemisphere BUT music in both (not necessarily true - new research) 1 Saturday, February 27, 2016 • overlap evidence: • BA 44 (part of Wernicke’s) activated in response to unexpected chords in music in music, activation is bilateral but only left in language • • posterior and anterior superior temporal cortex active in people who encounter unexpected chord/instrument • similar ERP patters to syntactic violations of language and harmonic deviations in music • theories of reconciliation: • how can neurological overlap exist alongside behavioral dissociation - relationship modular or shared • importance of comparative research • functional and neural architecture in both domains • role of different brain areas in hierarchical processing of complex sound sequences • broad principles of cognitive organization for complex info systems • shared syntactic resource integration hypothesis (SSIRH): language and music care a common set of processes that operate on different structural • representations • processing resources in frontal areas • representation regions in posterior brain areas • based on domain-specific cognitive theories of syntactic processing • comparative analysis best when driven by hypothesis, which requires consideration of empirically supported cognitive theory in both domains • dependency locality theory (DLT): • accounts for differences in complexity of grammatical sentences and for preferences in interpretation of syntactically ambiguous sentences • linguistic syntax requires: • structural storage: keeping track of predicted syntactic categories (expect verb after noun) • structural integration: connecting words to prior words • predicts that processing cost increases with distance between incoming elements and site of integration • processing cost = integration cost + storage cost • generates numerical predictions of syntactic processing • correspond to empirical data from reading time experiments • sentences presented and time taken to read each word • time taken to read each words corresponds with processing cost predicted by DLT • tonal pitch space theory (TPS): • both musicians and non musical have highly structured mental representation of musical pitch • quantifies tonal distance between 2 chords in musical sequence based on distance in structured cognitive space • “stable” pitches and chords perceived as closer to each other • separate keys perceived in orderly sets of distances from each other • numerical predictions of perceived tension • perceperception of tonal distance influences predicitions of the perceived ebb and flow of tension in musical sequences • tonal distance increases with tonal distance between chords • numerical predictions of TPS correspond to “tension profiling” scores produced by subjects who rate tension over time in a musical passage (listerners perceive musical elements in hierarchical relations) 2 Saturday, February 27, 2016 • convergence of domain-specific cognitive theories • structural integration (connecting x to y) is a key part os syntactic processing integration depends on distance between x and y in abstract cognitive space • • activatoin based framework • x —> y activates representation of x (incoming input) and reactivates T (prior input) • shared integration resource • processing resources serve to rapidly and selectively bring low-activation representations up to activation threshold for integration to occur neural areas and operations shared • • integration takes place in distinct regions • corrresponding neural overlap • not yet known • required within subjects comparative studies of language and music using localization techniques design domain-independent tasks with two distinct levels of syntactic integration • demands (quantified with DLT and TPS operations) • use fMRI to search for brain regions which show increased activation as a function of integration cost in both domains • existing evidence for shared integration resource • language processing research (Kaan & Swab, 2002; Haarmann & Kolk, 1991) processing regions in frontal areas provide resources for computations in posterior • areas where syntactic representations reside • MEG data (Koelsch, et al., 2000 & 2001) • activation of Broca’s area in harmonic processing • ERP component analysis (Patel, et al., 2007) • domain-independent tasks elicited statistically indistinguishable P600s • evidence for sidtinct representation regions • cases of music-specific syntactic deficits (Griffiths, et al., 1997; Peretz, 1993 & 1994) • associated with damage to superior temporal gyri • thought to be important in long-term representation of harmonic relations • congenital amusia (Ayotte, et al., 2002) • due to a problem with fine-grained pitch discrimination • suggests developmental failure to form cognitive representations of pitch • linguistic priming: how processing is influenced by a word’s syntactic or semantic relation to a passage • harmonic priming: influence of harmonic context on the processing of a target chord due to the acoustic similarity and the its distance in the cognitive construct • regions of activation • shared resources • anterior cingulate and medial frontal cortex • Domain general attention mechanisms (Slevc and Okada, 2014) • speech specific activation • bilateral anterior temporal gyrus • supports implication in combinatorial semantic processing, NOT hierarchical structures (Wong and Gallate, 2012) • conflicting activation in broca’s area • differences in activation based on tasks • passive listening • discrimination task 3 Saturday, February 27, 2016 • memory task • Music Task vs Passive Speech Listening Music discrimination – Passive speech listening • • No difference in activation of pars triangularis • Music memory – Passive speech listening • Greater activation of pars triangularis with music memory • reasoning for differences in activation • supporting Shared Synaptic Integration Resource Hypothesis overlapping activation seen when increased resources are need to complete the • speech or music task • Additional Support- cognitive control • Shared resources for linguistic and music processing in re-analyzing information and in conflict resolution (Slevc and Okada, 2014) • applications of SSIRH: improves treatments of speech disorders • Somatosensory Systems review figure 67c in Hendelman atlas • • modalities: • touch: discriminative, flutter vibration, crude touch • proprioception: where we are in space, muscle length and tension • temperature • nociception sensations of touch begin at the skin • • epidermis (outer layer) • dermis (inner layer) • mechanoreceptors • sense touch - mechxnosensitive ion channels • in organs - artery distension, stretch in digestive organs and bladder rapidly adapting: for discriminative touch • • fire at onset and offset of stimulus, not during • slowly adapting: for sensing steady pressure • fire the entire duration • types: • Pacinian corpuscles: in superficial subcutaneous layers, muscles, joints, internal organs • • end in nerve endings inside connective tissue • vibrations go through concentric layers to nerve endings to open mechanically gated ion channels • large receptive fields • rapidly adapting signal changes in pressure or sudden movement • • Meissner’s corpuscles: • superficial • inside connective tissue • 2 point discrimination • rapidly adapting 4 Saturday, February 27, 2016 • small receptive fields • Ruffini endings: • deep skin layers • enlarged nerve endings • sense steady skin pressure (stretch, tension) • slowly adapting • large receptive fields Merkel’s disks: • • superficial • enlarged nerve endings • sense steady pressure • slowly adapting • free nerve endings • respond to mechanical, thermal or noxious stimulation • generally for nociception • two point discrimination • higher density of mechanoreceptors (in hands and face, especially fingertips) • small receptive fields more cortical tissue devoted to these areas • • may be special mechanisms • lateral somatosensory tissue: info from hands and face small receptive field large receptive field fast adaptation Meissner’s corpuscle Pacinian corpuscle slow adaptation Merkel’s disk Ruffini’s ending • muscle spindle: stretch receptor • senses muscle length - proprioceptors are the LARGEST and FASTEST conducting axons • proprioceptor mechanoreceptors are in joints AND information from the muscle • Golgi tendon organs send info about tension on a muscle • sensory info comes from muscles, which then feeds info back into the motor system • primary afferent axons Aa (group I): • • highly myelinated • largest (13-20 um) and fastest (80-120 m/s) • proprioception in skeltl muscles (because you need to be able to react quickly to where your body is in space • transmit to spinal cord from proprioceptive system • NOT present in skin • AB (group II): • large and myelinated (not as big as Aa) • 30-70 m/s • mechanical stimulus Pacinian corpuscles • • touch info from SKIN (mechanoreceptors) • Ad (group III) • small, myelinated 5 Saturday, February 27, 2016 • 5-30 m/s • cold temperature fast pain information - sharp and localized • • C fibers (group IV) • small, unmyelinated • .5-2 m/s • mechanical stimulus - slow pain (dull aches) • temperature, itch dermatome: receptive fields - areas of the skin innervated by the left and right dorsal roots • • adjacent dorsal roots can innervate overlapping areas • demonstration: shingles • lesions or sores on skin activated on a specific nerve, so delineated within this area • pathways: • information sensed on skin, sent to spinal cord via fibers somatosensory info: • • from Pacinian corpuscles • less myelinated • cell bodies in dorsal root ganglion • information goes up spinal cord in dorsal columns to medulla and brainstem • collaterals also sent off for reflexive stimulation (in spinal cord) proprioceptive info • • highly myelinated • from muscle spindles • cell bodies in dorsal root ganglion • info goes up spinal cord in dorsal columns to medulla and brainstem • crosses in medulla • collaterals sent off for reflexive information • pain and temperature (via Ad and c fibers) • cell bodies in dorsal root ganglion • synapses in lamina of dorsal root • crosses immediately in dorsal horn and travels in contralateral side of spinal cord • spinothalamic pathway (aka anterolateral system) • somatotopic arrangement for information • lower extremities MEDIAL (Gracile fascicle) • sacral and lumbar • upper extremities lateral (cuneate fasciculus) • thoracic and cervical • in medulla, information synapses in gracile (lower) and cuneate nuclei (upper) • don’t have cuneate fascicles until thoracic level of spinal cord • cervical information enteres last • crosses midline in medulla, travels to thalamus • tactile pathway: • primary neurons from periphery (receptors) synapse in gracile or cuneate nucleus of medulla • dorsal columns in lower spinal cord are small, get bigger as you travel up the spinal cord • below medulla: dorsal columns - once above medulla pathway is called the medial lemniscus • secondary neuron: travels through pons and midbrain (brainstem) from medulla 6 Saturday, February 27, 2016 • synapses in ventroposterior lateral nucleus of thalamus • once at level of medial lemniscus, ventral in medulla at pons, lemniscus moves more dorsally again • • at level of midbrain, lemniscus is not more lateral • now medial lemniscus and anterolateral systems travel together • tertiary neuron: from thalamus to primary somatosensory cortex • anterolateral system (ALS): spinothalamic tract • travels to ventral posterolateral nucleus of thalamus (VPL) crossed immediately in spinal cord • • pain information DIRECTLY to VPL • reticulothalamic: information from neurons that project to reticular nuclei (medullary or pons reticular system) • synapse in intralaminar nuclei of thalamus (centromedian and parafascicular) • underlie alterting mechanism to painful stimuli involved in arousal and sleep/wake cycles • • different modalities processed on different parts of the gyrus: • area 3B: • primary somatic sensory • dense inputs from VP • very responsive to somatic input electrical stimulation evokes sensory experiences • • lesions impair somatic sensation • area 3A: sense of body position • area 1: input from 3B (texture) • area 2: size and shape • posterior parietal: polymodal information • most inputs terminate in layer IV - projections to other layers (stacked in columns) • homunculus: mapping of receptive fields of SI neurons produces orderly organization of information • info from specific body area terminates in specific brain region • columns of neurons devoted to one particular receptor type • cortical map plasticity: • surrounding areas invade cortical areas that lose innervation • ex: remove a finger and the cortical areas for the fingers around it expand to fill the area • larger cortical areas for greater stimulation in periphery • spinocerebellar tract: on both sides • proprioceptive information from periphery to cerebellum • posterior: lower limb and trunk • cuneocerebellar: upper limbs • synapse in lateral cuneate nucleus • anterior: touch info to cerebellum • crosses twice • rostral: upper limb 7 Saturday, February 27, 2016 Pain Systems Pain: an unpleasant sensory and emotional experience associated with actual or potential • tissue damage or described in terms of tissue damage, or both • types of nociceptors: • free nerve endings of unmyelinated c fibers and lightly myelinated Ad fibers • mechanical: strong pressure (crude touch) • thermal: extreme temperature chemical: histamines (itch) and others • • polymodal • hyperalgesia: increased sensitivity after injury • pain receptors: sensitize after injury • lower threshold and more responsive (more AP) • allodynia: perceiving innocuous stimuli as painful after tissue damage • involves vasodilation due to release of substance p • comparison of Ad and c fibers • first pain sensation registered by Ad axons (FAST) • pin prick • longer-lasting pain sensation mediated by c fibers (SLOW) • slow, dull, muscle pain • somatotopic organization: • lower extremities - lateral • upper extremities - medial • laminae in the dorsal horn • Ad into lamina I (posteromarginal nucleus) and V • c fibers into II (substantia gelatinosa) • III and IV - non noxious (nucleus proprius) • gate theory: • laminae that receive inputs from Ad and C fibers also get input from AB fibers • non-nociceptive fibers indirectly inhibit the effects of the pain fibers • this closes gate to transmission of stimuli (weakens signal sent to the thalamus) • anterolateral system (ALS): spinothalamic tract • travels to ventral posterolateral nucleus of thalamus (VPL) • crossed immediately in spinal cord • pain information DIRECTLY to VPL • reticulothalamic: information from neurons that project to reticular nuclei (medullary or pons reticular system) • synapse in intralaminar nuclei of thalamus (centromedian and parafascicular) • underlie alterting mechanism to painful stimuli • involved in arousal and sleep/wake cycles • therapies for intractable pain • thalamic lesioning (CM-PF) • deep brain stimulation • descending pain control: can selectively modulate pain in different areas • regulated by emotion and behavioral state • somatosensory cortex • insula and anterior cingulate • periventricular nucleus • periaqueductal gray (stimulation causes analgesia) 8 Saturday, February 27, 2016 • raphe nuclei • pain perception involves cognitive processes cortico-limbic striatal circuits: leads to autonomic reactivity • • cognitive appraisal • emotional reaction • behavioral response attention • pain neuromatrix: leads to thalamocortical relays • ACC, insula, PFC. amygdala, hypothalamus, sensory cortex leads to inflammatory and bimolecular mediators • • also leads to autonomic reactivity • can also control descending pain modulation • noxious stimulus leads to stimulation of nociceptor —> dorsal horn of the spine • leads to thalamocortical relays • activation of medial prefrontal cortex (mPFC) and posterior cingulate cortex (PCC) in pain rumination 9

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Chapter 13, Problem 4 is Solved
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Textbook: Calculus
Edition: 8
Author: James Stewart
ISBN: 9781285740621

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(a) A projectile is fired from the origin down an inclined