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Answer: Find all rational, irrational, and complex zeros

Precalculus: Mathematics for Calculus | 6th Edition | ISBN: 9780840068071 | Authors: James Stewart ISBN: 9780840068071 161

Solution for problem 63 Chapter 3

Precalculus: Mathematics for Calculus | 6th Edition

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Precalculus: Mathematics for Calculus | 6th Edition | ISBN: 9780840068071 | Authors: James Stewart

Precalculus: Mathematics for Calculus | 6th Edition

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Problem 63

Find all rational, irrational, and complex zeros (and state their multiplicities). Use Descartes Rule of Signs, the Upper and Lower Bounds Theorem, the Quadratic Formula, or other factoring techniques to help you whenever possible.P1x2 x 4 6x 3 17x 2 28x 20

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Tuesday, April 19, 2016 Week 14 Brainstem • ventral view: • optic chiasm can help you localize • diencephalon above that oculomotor nerve comes from midbrain • • Edinger-Westphal nucleus comes from oculomotor nerve - parasympathetic function • oculomotor nucleus also • cerebral peduncles: bulbous parts of pons • medullary pontine function: abducens • medulla: facial nerve • • nucleus deep in pons • nerve wraps up and around abducens nucleus (fiber bump) • trigeminal nerve: sensory and motor function to face • large stalk coming off pons • vagus: parasympathetic, sensory and motor hypoglossal nerve • • lower medulla slice: (67c) • inferior olive: squiggly structure • connection to cerebellum • sensory info • medial lemniscus: part of dorsal columns proprioception and touch info • • in midline at medulla level • spinothalamic tract: lateral • pain and temperature • as you go through the spinal cord, travels closer to medial lemniscus • nucleus gracilis: somatosensory from lower body medial • • nucleus cuneatus: upper body • lateral • spinal nucleus and tract • solitary nucleus and tract • sensory information from organs nucleus just below nucleus gracilis and cognates • • looks like a bullseye • reticular: medial • motor: • corticospinal tract in medullary pyramids • bottom of medulla (pyramids) vagus nerve (X) • • visceral motor • dorsal motor nucleus - medial • nucleus ambiguus - hard to see on sections, very deep within medulla • visceral motor 1 Tuesday, April 19, 2016 • hypoglossal: somatic efferent info • controls tongue muscles medial longitudinal fasciculus: sensory and motor info between nuclei for head, neck • and eye movement • upper medulla: (67A) • inferior olive • cereellar peduncle • sensory info: medial lemniscus: large (all fibers have crossed) • • anterolateral/spinothalamic: moving more medially to travel with dorsal columns • spinal trigeminal nucleus: fairly lateral • solitary nucleus • vestibular nuclei • cochlear nuclei reticular system • • motor • corticospinal tract • medial longitudinal fasciculus • 4th ventricle • can see cerebellum lower pons: (66C) • • sensory: • superior olive: auditory information processing (inter-aural timing differences) • trapezoid bodies: interaural time difference and communication between superior olive and nuclei • medial lemniscus • spinothalamic • nuclei: • facial - fibers exit in cranial nerve 7 and wrap around abducens nucleus • form facial colliculus (bumps) • vestibular nuclei at junction of pons and medulla • spinal trigeminal (goes from midbrain to medulla) • cranial nerve 8 • motor information • corticospinal • cranial nerve 7 • abducens nucleus and abducens • MLF • upper pons: (66B) • snowman structure: bulbous • pontine nuclei • corticospinal tract • trochlear nerve (4) • sensory • medial lemniscus: now lies laterally • anterolateral system • reticular formation • locus ceruleus: addiction, stress response • superior cerebellar peduncle 2 Tuesday, April 19, 2016 • 4th ventricle (small in upper pons) • lower midbrain landmarks: • • colliculi (dorsally) • fibers (ventrally) • fronto-pontine connections • sensory info from temporal parietal and occipital lobe to pons • substantia nigra - dark due to melanin in cells decussation of cerebellar peduncles • • medial lemniscus and anterolateral now more lateral (together) • lateral lemniscus: auditory info • reticular formation • cerebral aqueduct • periaqueductal gray sensory: • • mesencephalic nucleus of trigeminal nerve • motor info: • MLF • trochlear nucleus • upper midbrain: red nucleus • • oculomotor nucleus/nerve • substantia nigra Cranial Nerves • most nerves come from the brainstem except for olfactory and optic nerve (part of CNS) • functional divisions • sensory: afferent • more continuous than motor nuclei SA: special afferent • • olfactory and optic nerve (slightly different function) • GVA: general visceral afferent • sensory info from organs • GSA: general somatic afferent • normal sensory info trigeminal • • vagus • motor: efferent • SVE: special visceral efferent • from brachial endoderm: face and neck • innervation to muscles of mastication and larynx muscles mandibular nerve (root of trigeminal) • • facial nerve • spinal accessory • GVE: general visceral efferent • mainly parasympathetic function for cranial nerves • vagus 3 Tuesday, April 19, 2016 • facial • oculomotor glossopharyngeal • • GSE: general somatic efferent • oculomotor • trochlear • abducens • medial —> lateral GSE • • GVE • SVE • SVA • GVA • SSA GSA • • medial: innervate skeletal muscle • midbrain nuclei: • oculomotor (GSE) • Edinger-Westphal (GVE) • trochlear (GSE) mesencephalic nucleus and tract of V (GSA) • • pons • abducens (GSE) • superior salivatory (GVE) • motor trigeminal (SVE) • motor facial (SVE) • principle sensory (GSA) • spinal trigeminal (GSA) • medulla • hypoglossal (GSE) • dorsal motor (GVE) • inferior salivatory (GVE) • nucleus ambiguus (SVE) • solitary • gustatory (SVA) • cardiorespiratory (GVA) • vestibular (SSA) • cochlear (SSA) • spinal trigeminal (GSA) • sensory info goes through brainstem, synapses in sensory nuclei • motor pathways start in M1 or frontal eye fields, bilateral innervation to lower motor neuron, which projects out onto cranial nerve • nerves: • 1. olfactory - sensory • smell • projection from olfactory cells to bulb • bulb carries info back to the cortex - only sensory system that doesn’t synapse in thalamus before cortex • 2. optic - sensory 4 Tuesday, April 19, 2016 • vision • projects to LGN and superior colliculus LGN projects to V1 • • 3. oculomotor - motor • GSE - somatic function: • innervates 4 of the 6 eye movement muscles • ipsilateral inferior rectus muscle - bottom of the eye • ipsilateral inferior oblique muscle - turning eye ipsilateral medial rectus muscle - nasal portion of eye • • contralateral superior rectus muscle - top of eye • levator palpebrae - elevation of eyelids • innervation to ciliary and pupillary constrictor muscles • lies closest to midline • GVE - parasympathetic function starts in Edinger-Westphal nucleus • • synapses in ciliary ganglia, which innervate ciliary muscles • just lateral to somatic function • starts in junction of midbrain and pons • somatic motor function starts in oculomotor nucleus, goes to eye muscles • 4. trochlear - motor smallest cranial nerve • • most medial • GSE - somatic function • innervates contralateral superior oblique muscle (acts as a pulley for the eye) • rotates eye inward • lies just caudal to oculomotor nucleus • exits brainste on dorsal side • ALL fibers are crossed • 5. trigeminal - both • largest nerve (stalk) • arises from pons • most somatosensory info from face • motor function: • SVE: muscles of mastication/larynx muscles • motor nucleus of V • most lateral of the motor nuclei • sensory function: • GSA: somatosensory pain and temperature info from face • ophthalmic nerve - forehead/upper eyelid skin • maxillary nerve - cheek and lower eyelids • mandibular - lateral scalp, ears, lips, cheeks • nuclei: • mesencephalic - proprioceptive info • main sensory - touch from face • spinal nucleus of V - pain afferent • most lateral nuclei • 6. abducens - motor • starts in junction of pons and medulla • most medial 5 Tuesday, April 19, 2016 • GSE: • innervate ipsilateral lateral rectus muscle (contraction abducts eye OUTWARD/ laterally) • temporal side of the eye • 7. facial - both • from junction of pons and medulla • SVE: branchial muscles of facial expression • muscles making you smile, frown, etc elevation of eyelid • • motor facial nucleus • pons • GVE: parasympathetic • arises in brainstem, synapses in ganglion and ends in glands of nose and mouth • lacrimal gland and mucous membranes of the nose - pterygopalatine ganglion sublingual and submandibular glands (mouth)- submandibular ganglion • • superior salivatory nucleus (input from hypothalamus) • pons • SA: • inner side of external ear - geniculate ganglion to spinal trigeminal nucleus • long nucleus from pons to medulla solitary tract nucleus - taste to anterior 2/3 tongue • • projects to thalamus • GVA: • nasal/oral sensation - geniculate ganglion • solitary tract nucleus • projects to thalamus • bilateral and contralateral innervation • facial nerve carries info from BOTH upper motor neurons (bilateral) • lower parts of the face get contralateral info (smiling muscles, etc.) • upper muscle of the face get bilateral innervation • lower motor neurons travel ipsilaterally, upper motor neurons travel • not necessarily receptors on tongue that detect taste, but function of the cranial nerves • lesion to lower motor neuron - lesion to facial nerve • lose innervation to lower AND upper part of half of face —> Bell’s palsy • lesion to upper motor neuron: only lose innervation to contralateral, lower side of the face • upper face gets bilateral innervation, so it’s still okay • 8. auditory-vestibular - sensory • aka vestibulocochlear • arises laterally from pons/medullary junction • SA: hearing and balance • Scarpa (vestibular) ganglion to vestibular nuclei • vestibular info from semicircular canal • spiral (cochlear) ganglion to cochlear nuclei • auditory info from cochlea • middle and most lateral of the afferents • 9. glossopharyngeal - both • parasympathetic • SVE: 6 Tuesday, April 19, 2016 • to stylopharyngeus muscle (elevates pharynx during swallowing and speech) • nucleus ambiguus (in medulla) parasympathetic • • GVE: • to parotid gland via otic ganglion • inferior salivatory nucleus • GVA: • from carotid body and sinus senses oxygen tension baroreceptors • • nucleus of solitary tract • GSA: • sensation from back 1/3 of tongue • skin of external ear, inner tympanic membrane, gag reflex • nucleus of spinal trigeminal SA: • • taste from posterior 1/3 of tongue • solitary tract nucleus (gustatory portion) • 10. vagus - both • arises from medulla • parasympathetic GVE: • • smooth muscle and glands of pharynx, larynx, esophagus, thoracic and abdominal viscera • cardio-respiratory visceral efferents • dorsal motor nucleus of vagus • parasympathetic • SVE: • to striated larynx muscles (some tongue and pharynx) • nucleus ambiguus • GVA: • base of tongue, epiglottis, larynx, esophagus, thoracic and abdominal visceral • stretch/chemo receptors in aortic arch • nucleus of solitary tract • SA: • taste to base of tongue • nucleus of solitary tract • external ear/external tympanic membrane • spinal trigeminal nucleus • 11. spinal accessory - motor • arises from border or medulla and spinal cord • lower motor neuron travels ipsilaterally • GSE: • spinal roots to sternocleidomastoid and trapezius - head, neck and shoulder movement • 12.hypoglossal - motor • arises from upper medulla (right on the edge of the medullary pyramids) • hypoglossal nucleus • GSE: • most tongue muscles (except palatoglossus, which is controlled by vagus) 7 Tuesday, April 19, 2016 • ipsilateral only (lower motor neuron) • lower motor neurons travel ipsilaterally upper motor neurons travel bilaterally • • NTS: • gustatory information - upper portion • cardiorespiratory sensory information - lower part • eye movement • medial rectus: on nasal portion adduction (bring eye inward) • • lateral rectus: on temporal portion • brings eye outward - abduction • inferior rectus: move eye down • superior rectus: elevates eye • inferior oblique: axis - up and out superior oblique: turns eye on axis (down and in) • • extra ocular eye muscles: • SO4LR6 • superior oblique - 4 • lateral rectus - 6 • everything else - oculomotor nerve motor info: medial • • closer to sulcus limitans: visceral • sensory info: lateral • visceral most medial • as you go lateral, more somatic 8

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Chapter 3, Problem 63 is Solved
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Textbook: Precalculus: Mathematics for Calculus
Edition: 6
Author: James Stewart
ISBN: 9780840068071

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Answer: Find all rational, irrational, and complex zeros