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In each of 1 through 10, find a fundamental matrix for the

Advanced Engineering Mathematics | 7th Edition | ISBN: 9781111427412 | Authors: Peter V. O'Neill ISBN: 9781111427412 173

Solution for problem 10.6 Chapter 10

Advanced Engineering Mathematics | 7th Edition

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Advanced Engineering Mathematics | 7th Edition | ISBN: 9781111427412 | Authors: Peter V. O'Neill

Advanced Engineering Mathematics | 7th Edition

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Problem 10.6

In each of 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 3x1, x 2 = 5x1 4x2

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Psychology Exam #2 Study Guide- Professor Cate’s Class Highlight = Important People Highlight = Important Idea Highlight = Key Term CHAPTER 6: SENSATION AND PERCEPTION  Basic Principles of Sensation and Perception  Your brain will interpret, perceive these topics as they enter you sense organs:  Sensation vs. Perception, Bottom-Up vs. Top-Down Processing  Transduction and Thresholds  Sensory Adaption  Perceptual Set  Context Effects on perception  Emotion/Motivation effects  Sensation vs. Perception  Sensation: “The process by which our sensory receptors and nervous system receive and represent stimulus energies from our environment.  The brain receives input from the sensory organs (sends info to brain).  Perception: “The process of organizing and interpreting sensory information, enabling us to recognize meaningful objects and events.”  The brain makes sense out of the input from sensory organs.  Making Sense of the World  Bottom-up processing: taking sensory information and then assembling and integrating it.  “What am I seeing”  Top-down processing: using models, ideas, and expectations to interpret sensory information.  “Is that something I’ve seen before”  Your expectations determine what the ambiguous image is. (For example: the painting inside the bottle, the kids would see dolphins but adults would see sexual images.)  Sensory Adaptation  To help detect novelty in our surroundings, our senses tune out a constant stimulus, such as:  A rock in your shoe  The ticking of a clock  Visual sensory adaption will be tested when discussing opponent-process theory.  Perceptual Set  Perceptual set is what we expect to see, which influences what we do see.  An example of top-down processing  Picture of Loch Ness monster or a tree branch Flying saucers or clouds  Perceptual set can be “primed” (ex: picture of an old woman but also young lady)  Context Effect of Perception  Perception of size depends on context and surroundings.  Spelling test answers:  Double pear  Apple payor  Payee pair  You would choose the word based on context.  Effect of Emotion, Physical State, and Motivation on Perception:  Experiments show that:  Destinations seem farther when you’re tired.  A target looks farther when your crossbow is heavier.  A hill looks steeper with a heavy backpack, sad music, or walking alone.  Something you desire looks closer.  Conclusion: Our estimates are more optimistic when we don’t have to do the work or it doesn’t affect us.  Vision, and Perceptual Organization and Interpretation  Topics:  Vision (sensation):  The eye  From light input to mental images  Retina and receptors  Feature detection  Parallel processing  Color vision  Visual Organization:  Form, depth, and motion perception  Size, shape, and color constancy  Visual interpretation  Restored vision  Perceptual adaptation  ESP, perception without sensation  Vision: Energy, Sensation, and Perception  We encounter waves of electromagnetic radiation.  Our eyes respond to some of these waves.  Our brain turns these energy wave sensations into colors.  The Retina: **What the retina does with light: The white arrows show light going in, triggering a reaction in the rods and cones. The black arrows show neural impulses generated in the bipolar cells and gathered by the ganglion cells to be funneled into the optic nerve “pipeline” of nearly one million ganglion cell axons. That’s a lot of potential information that can be shipped to the brain.  The Blind Spot:  Blind spot: an area of missing info in our field of vision.  Occurs because the eye has no receptor ells at the place where the optic nerve leaves the eye.  Test: one eye closed and the other eye fixed on an object (one object will disappear)  Photoreceptors: Rods and Cones  When light reaches the back of retina, it triggers chemical changes in two types of receptor cells:  Rods help us see the black and white actions in our peripheral view and in the dark. (green)  Cones help us see sharp colorful details in bright light. (blue)  Visual Information Processing:  The rods and cones send messages to ganglion and bipolar cells and on to the optic nerve.  Once neural signals enter the optic nerve, they are sent through the thalamus to the visual cortex.  The images we “see” are not made of light; they are made of neural signals, which can be produced even by pressure on the eyeball. (people say that they “see stars” when they get punched hard)  Turning Neural Signals into Images:  In the visual cortex are neurons called feature detectors: they respond to certain visual aspects of the environment.  These cells in turn send information to neural networks (supercell clusters) that can perform tasks such as recognizing individual faces.  People who have damage in these areas of the brain have it difficult to see certain kinds of objects and integrate them into a whole. (Ex: recognize the eyes, nose, mouth, etc. but cannot recognize a face when they are put together)  Parallel Processing:  Turning light into the mental act of seeing:  Light chemical reactions neural impulses features objects  One more step…  Parallel Processing: building perceptions out of sensory details processed simultaneously in different areas of the brain.  Visual Processing:  Scene  Retinal processing: receptor rods and cones bipolar cells ganglion cells  Feature detection: brain’s detector cells respond to specific features—edges, lines, and angles  Parallel processing: brain cell teams process combined information about color, movement, form, and depth  Recognition: brain interprets the constructed image based on information from stored images  Color Vision:  Young-Helmholtz Trichromatic (Three-color) Theory  According to this theory, there are three types of color receptor cones—red, green, and blue. All the colors we perceive are created by light waves stimulating combinations of these cones.  For example: you could see the color of an orange because it absorbs all light except the wavelength that our brain interprets as orange. Notes that the red, green and blue don’t actually refer to the appearance of the cones; they are the colors to which these three cones react.  Color Blindness:  People missing red cones or green cones have trouble differentiating red from green trouble reading the numbers. **Lighting makes a difference when seeing colors too.  Opponent-process theory refers to the neural process of perceiving black; similarly, yellow vs. blue, and red vs. green are opponent processes.  For example: dogs have black and white vision because they are lacking red receptors, so their vision has simpler color perception, dichromatic, not monochromatic.  Turning light waves into mental images/movies… Visual Perceptual Organization  We have perceptual processes for enabling us to organize perceived colors and lines into objects:  Grouping incomplete parts into gestalt wholes.  Seeing figures standing out against background.  Perceiving form and depth  Keeping a sense of shape, size, and color constancy despite changes in visual info  Using experience to guide visual interpretation  Restored vision and sensory restriction  Perceptual adaptation  The Role of Perception: Our senses take in the blue info on the right. However, our perceptual processes turn this into:  A white paper with blue circle dots, with a cube floating in front.  A white paper with blue circle holes, through which you can see a cube.  A cube sticking out to the top left, or bottom right.  Blue dots with angled lines, but cannot see the cube.  Figure- Ground Perception:  In most visual scenes, we pick out objects and figures, standing out against a background.  Some art muddles this ability by giving us two equal choices about what is figure and what is “ground”: Goblet or two faces Stepping man, or arrows  Grouping: How We make Gestalts  “Gestalt” refers to a meaningful pattern/configuration forming a “whole” that is more than the sum of its parts.  Three of the ways we group visual info into “whole” are proximity, continuity, and closure.  Example: 3D dog house **Closure and continuity are part of what makes the boards look like they continue behind other boards and connect in places they really don’t, even though these connections and closures create an image of a house that violates our sense of how things can work in 3D space.  Visual Cliff: A test of Depth Perception  Babies develop this ability at crawling age more mobility, more fear.  Perceiving Depth: Binocular Methods  Unlike other animals, humans have two eyes in the front of our head. This gives us retinal disparity; the two eyes have slightly different views. The more different the views are, the closer the object must be.  This is used in 3D movies to create the illusion of depth, as each eye gets a different view of “close” objects.  How do we perceive depth from a 2D image... by using monocular (needing only one eye) cues.  Monocular Cue: Interposition  Interposition: When one object appears to block the view of another, we assume that the blocking object is in a position between our eyes and the blocked object.  Monocular Cue: Relative Size  We intuitively know to interpret familiar objects (of known size) as farther away when they appear smaller.  Depth perception  Monocular Cues: Linear Perspective and Interposition  The flowers in the distance seem farther away because the rows converge. Our brain reads this as a sign of distance.  Monocular Cue: Relative Height  We tend to perceive the higher part of a scene as farther away.  This scene can look like layers of buildings, with the highest part of the sky. If we flip the picture, then the black part can seem like night sky because it is now the highest in the picture.  Monocular Cues: Shading Effects  Shading helps our perception of depth.  Does the middle circle bulge out or curve inward  Light and shadow creates depth cues.  Monocular Cues: Relative Motion  When we are moving, we can tell which objects are farther away because it takes longer to pass them.  A picture of a moon on a sign would zip behind us, but the actual moon is too far for us to pass.  Perceptual Constancy:  Our ability to see objects as appearing the same even under different lighting conditions, at different distances and angles, is called perceptual constancy (top-down process).  Examples: color and brightness constancy/ shape and size constancy  Color Constancy:  This ability to see a consistent color in changing illumination helps us see illumination helps us see the three sides as all being yellow, because our brain compensates for shading.  As a result, we interpret three same-color blue dots, with shades that are not adjusted for shading, as being of three different colors.  Brightness Constancy:  On this screen, squares A and B are exactly the same shade of gray. You can see this when you connect them. So why does B look lighter  Explain that our brains compensate for shadows and other context by perceiving a constant color shade/brightness even when things are in shadow. This means mentally erasing the shadow to see objects in a lighter shade. This process, plus the checkerboard context, makes B seem lighter to our brain than the images sensed by our eyes.  Shape Constancy:  Shape constancy refers to the ability to perceive objects as having a constant shape despite receiving different sensory images. This helps us see the door as a rectangle as it opens. Because of this, we may think the red shapes on screen are also rectangles.  Size Constancy:  We have an ability to use distance-related context cues to help us see objects as the same size even if the image on the retina becomes smaller.  The Ames room was invented by American ophthalmologists Adelbert Ames, Jr. in 1934.  The Ames room was designed to manipulate distance cues to make two same-sized girls appear very different in size.  Visual Interpretation: Restored vision, sensory restriction (Experience shapes our visual perception)  People have grown up without vision but then have surgically gained sight in adulthood. They learned to interpret depth, motion, and figure-ground distinctions, but could not differentiate shapes or even faces.  Animals raised at an early age with restrictions, e.g. without seeing horizontal lines, later seem unable to learn to perceive such lines.  We must practice our perception skills during a critical period of development, or these skills may not develop.  Perceptual Adaptation:  After our sensory information is distorted, such as by a new pair of glasses or by delayed audio on a television, humans may at first be disoriented but can learn to adjust and function.  This man could learn eventually to fly an airplane wearing these unusual goggles, but here at first at first, he is disoriented by having his world turned upside down.  The Nonvisual Senses:  There’s more to Sensation and Perception than meets the eye.  Hearing: From sound to ear to perceiving pitch and locating sounds.  Touch and Pain sensation and perception  Taste and Smell  Perception of Body Position and Movement  Mixing the different sense together:  Sensory interaction occurs when different sense influence each other.  For example:  A burst of sound makes a dim light source more visible.  Flavor is an experience not only of taste, but also of smell and texture.  Seeing text or lip movement, or even feeling the puff of air from consonants, affects what words we hear.  Synaesthesia: a condition when perception in one sense is triggered by a sensation in a DIFFERENT sense.  Some people experience synaesthesia all the time, reporting that, “the number 7 gives me a salty taste” or “rock music seems purple”.  Embodied Cognition:  It’s no coincidence that we use sensation words to describe feelings. Studies seem to show that:  Holding a warm mug promotes social warmth  Social rejection looks like pain reception in the brain.  Words on a heavy clipboard seem weighty.  Being ignored (cold shoulder) makes a room seem colder.  Leaning left physically leaning left politically.  In a foul smelling room, people were more likely to suspect bad intentions (foul play) by others.  Embodied Cognition refers to the effect of body experience on feelings, attitudes, thought, and judgement.  Extrasensory Perception (ESP):  Extrasensory perception can defined, literally, as perception without sensation.  Believers in ESP think that it involves getting accurate information directly to the mind, skipping the know senses.  Types of ESP:  Telepathy: “reading” messages from other minds  Clairvoyance: “seeking” remote events  Precognition: “knowing” the future  The evidence for ESP is anecdotal and controversial; people seem to notice times when predictions come true and perceptions match reality, but tend to disregard the times when they do not.  Summarizing the Senses:

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Chapter 10, Problem 10.6 is Solved
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Textbook: Advanced Engineering Mathematics
Edition: 7
Author: Peter V. O'Neill
ISBN: 9781111427412

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