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Use indirect reasoning to prove that log2 5 is an

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

Solution for problem 14 Chapter 1

Calculus | 8th Edition

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

Calculus | 8th Edition

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Problem 14

Use indirect reasoning to prove that log2 5 is an irrational number

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Behavioral Neuroscience Study Guide – Exam 3 *Study Tip: When studying for this class exams you should overview ALL of your notes in a good amount. It is good to focus on the more complicated concepts but do not neglect the simpler concepts because there will most likely be questions on it as well. This guide focuses on explaining on the more complicated concepts but you should not study this guide alone. Overview of Neuroscience Methods: Animal Methods - Patch Clamping: Micropipette is used to seal off a piece of the neuronal membrane - Immunocytochemistry: “programming” antibodies to detect specific proteins - Microdialysis: Extracts ECF fluid from inside the brain to the outside through a pump - Tracing: Labels neurons (via dye, autoradiography, virus etc…)so you can follow their pathways - Optogenetic: Alters neurons to be light sensitive so it can be triggered with different Knockout wavlengths - Ablation: Destroying of tissue - Single-cell recordings: Placing an electrode into a brain area and passively record what occurs - Electrical Stimulation: Place an electrode in the brain area and use it to stimulate - Knockouts: Removing a gene from an animal and noting differences Human methods - Patients: Compare populations with brain damage V. those without brain damage - Electrocorticography (ECoG): Implanted across the frontal cortex, can locate activity (i.e. seizure sources) - Electrical brain Stimulation (EBS): Stimulating the brain with the ECoG - Computerized Tomography (CT): Combines 2-D x-rays to create 3-D image of the brain structure - Positron Emission Tomography (PET): Functional 3D images measured by brains consumption of radioactive glucose - Magnetic Resonance Imaging (MRI): Magnetic fields and radio frequency’s used for structural and functional 3D brain imaging - Functional MRI (fmri): When the MRI is set to measure function, not structure. Measures blood oxygenation - Electroencephalography (EEG): Places electrodes on the scalp to measure electrical activity - Magnetoencephalography: Measures magnetic fields generated by the brain - Transcranial Magnetic Stimulation (TMS): Disrupts cortical activity with a magnetic pulse, causing a brief error - Behavioral Genetics: Attempts to correlate behavior with similarity/variability in the genome Overview Vision: Stimulus - Light (Wavelength, Amplitude) = Electromagnetic energy) o Saturation = purity of color The Sensory Apparatus - The Eye - The Retina Neural Pathways - Retinohypothalamic : Retina to hypothalamus - Retinopretectal: Retina to pretectum - Retinocollicular: Retina to superior colliculus - Retinogeniculostriiate: retina to LGN o Handles all detailed and conscious vision o RetinaLGNV1 Cortical Representations - Area V1 o Ventral Stream (When) o Dorsal stream (Where) Stimulus Humans can detect light between a wavelength of 380nm – 760 nm - Amplitude = Affects brightness - Wavelength = Affects color  Perception is further affected by: o Hue o Saturation o Brightness Sensory Apparatus  Light enters the eye and hits the retina (in the back) then proceeds to go through its 3 layers:  The retina layers have Laminar organization and function BACKWARDS from their anatomical order o Light passes through the ganglion layer  bipolar layer photoreceptor layer because they are transparent. o Once the light hits the photoreceptor layer it triggers the information processing and works its way BACKWARDS through the layers. The ganglion cells connect to the optic nerve which then send the information to the brain Retina layers (in functional order) 1. Photoreceptor layer: Transduces light into neural signals (forms synapse with bipolar cells) a. Rods : Night Vision b. Cones: Detailed/color vision i. S-Cones ii. M-Cones iii. L- cones 2. Bipolar cells: Transfer information from the photoreceptors to ganglion cells 3. Ganglion Cells: Project to the brain through the optic nerve RODS V. CONES CONES RODS Most concentrated in the Fovea (central retina) Most Concentrated in the Peripheral retina (not present in the fovea) Sensitive to moderate – high levels of light Sensitive to Low levels of light Provide Information about hue Provide only monochromatic information Provides excellent acuity Provides poor acuity Visual Pathways =Contralateral Organization Eye: Has both Visual Hemifields (Right and left) Optic Tract: Keeps track of the visual hemifield on its side Optic chiasm: Contains both sides of space (loss = blindness) Receptive fields of Ganglion Cells On Center Cells: Pick up light stimulus on a dark background Off Center cells: Pick up dark stimulus on a light background Cortical Representations 1. Area V1: Retinotropic organization ( and don’t forget contralateral) pictured in the image below V1 Receptive fields o Binocularity: Refers to a neuron having a receptive field in each eye o Orientation selectivity : Responds preferentially to edges with a particular orientation o Direction selectivity: receptive fields respond preferentially to a stimulus with a particular orientation moving in a particular direction o Color selectivity: receptive fields respond preferentially to particular color. V1 color selective cells exhibit response properties much more complex than those in the retina or LGN 2. Ventral Streams = What i. Lateral Occipital Cortex (LOC) ii.Fusiform Face Area (FFA) iii.Parahippocampal Place Area (PPA) o Damage causes visual agnosia – The inability to identify objects visually 3. Dorsal Streams = Where i. Middle Temporal Area: Involved in the processing of motion ii. Posterior Parietal Cortex (PPC): Consists of several subareas that represent space as it pertains to the orienting of attention, planning and compensation of eye movements, and visual control of limbs Damage o Akinetopsia: an inability to perceive motion o caused by damage to middle temporal area) o Optic Ataxia: –failure to generate visually guided movements o Caused by damage to PPC o Neglect Syndrome: Neurological disorder that occurs after damage to PPC o Typically the right posterior parietal is damaged  they would neglect the left side of space  Example: If they were to draw a clock they would probably draw only the right side of the clock (12->6)  Ignore the left side of space on a large and small/detailed scale (ex: in the clock example above maybe they would only draw the right portion of the 6) EXAMPLE QUESTIONS FROM CLASS 1. How could you determine which brain areas the Inferior Colliculus projects to 2. What method would you use to measure the electrical current generated by opening a single nicotinic receptor 3. What method would you use to understand what type of objects a neuron in visual cortex reports 4. What method would you use to determine the role of D2 receptors on long-term memory 5. A patient in the ER was accidently shot with a nail gun. It is suspected that the nail is lodged in the patient’s brain. What type of structural brain scan should be performed Why 6. What method would you use to measure electrical signals generated during a seizure 7. What method would you use to measure the volume of the hippocampus in the healthy human brain 8. What method would you use to stimulate cortical tissue in the healthy human brain 9. The optic nerve is made up of axons originating from____ 10. What Type of photoreceptor are you using to see during the day 11. How would damage to the right optic tract affect vision 12. Damage to the left lateral geniculate nucleus would lead to: A. Complete blindness B. Blindness in the right visual field C. Blindness in the Left Visual Field 13. Bilateral damage to the primary visual cortex would lead to D. Complete blindness E. Blindness in the right visual field F. Blindness in the Left Visual Field Answers to EXAMPLE QUESTIONS from Class 1. Answer: Anterograde Tracing (with dye ) 2. Answer: Patch clamping 3. Answer: Single-cell recording 4. Answer: Knockout method (use a mouse that has the D2 receptor turned off and compare it to a similar mouse with the D2 receptor still in tact) 5. Answer: CT Scan, MRI would be dangerous since it would cause the nail to move. CT scan will provide the next best imaging besides MRI 6. ANSWER: EEG 7. Answer: MRI (it is the ideal way to see an image) 8. Answer: TMS 9. Answer: Ganglion cells 10. Answer: Cones 11. Answer: You would lose the left field 12. B. Blindness in the right visual field 13. D. Complete Blindness Answers to PRACTICE questions From Blackboard 1. INCREASE in firing rate (turned on) 2. Decreases firing rate from base line 3. Remains at Base line 4. Remains at Base line 5. Remains at Base line Explanation of Answers 1-5: This is an Off-Center cell and the black dots and lines represent darkness while the background is light. Recall in the notes (week 10) that Off- center cells pick up dark stimulus on a light background. Dark spots in the center stimulate it while dark spots in the surround inhibit it. - In 3-5 there is equal parts darkness going through the center and surround so there is no OVERALL simulation or inhibition 6. C. 2 7. A. inject a retrograde tracer into the inferior colliculus 8. A. single-cell recordings from the hippocampus 9. A. Fovea 10. Complete damage to left V1 will cause right hemifield blindness. Bilateral damage to the FFA will give patient the inability to recognize faces 11. Adding a second type of rod would give you color vision at night because you could differentially represent wavelengths 12. The FFA would become the “wolf-face” area and they would be bad at recognizing human faces 13. Allows you to pick up contrast by balancing out/cancellig 14. Left LGN damage will cause right visual field blindness. This type of V1 damage would cause them to lose left peripheral field

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Chapter 1, Problem 14 is Solved
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Textbook: Calculus
Edition: 8
Author: James Stewart
ISBN: 9781285740621

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Use indirect reasoning to prove that log2 5 is an