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Solved: In Exercises 47 and 48, use a Taylor series to

Calculus: Early Transcendental Functions | 6th Edition | ISBN: 9781285774770 | Authors: Ron Larson ISBN: 9781285774770 141

Solution for problem 48 Chapter 16

Calculus: Early Transcendental Functions | 6th Edition

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Calculus: Early Transcendental Functions | 6th Edition | ISBN: 9781285774770 | Authors: Ron Larson

Calculus: Early Transcendental Functions | 6th Edition

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Problem 48

In Exercises 47 and 48, use a Taylor series to find the first terms of the series solution of the differential equation under the specified initial conditions. Use this polynomial to approximate for the given value of x.

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Chapter 1: Introduction to biology of Coral Reefs  Coral reefs are the largest structure ever built by living organisms o Great diversity of organisms in coral reefs o Every organism is linked to many others in complex web of interactions  Extremely vulnerable to destruction by human activities  Corals protect coastlines from damage by storm waves and shelter fish/seafood  Economic benefits o Building o Decorative o Pharmaceuticals o Diving tourism Where are coral reefs Corals require o Warm waters, 25 degrees north-25 degrees south o Require sunlight, so shallow bottom and clean water o Normal salinity o Moderate water movement  Found along tropical and subtropical coastlines in >100 countries What are corals  Corals are part animal, part plant and part rock o Rock: as corals grow, they deposit skeletons of calcium  Live tissue is thin layer on surface  Lays calcium carbonate down through growing o Animal: “cnidarians”, related to sea anemones and jellyfish  All cnidarians have a mouth ringed by tentacles and a simple sac-like gut  Sac with mouth down is a medusa  Sac with mouth up is a polyp  Nematocytes: cnidarian tentacles are armed with stinging tentacles for defense and prey capture o Plant: most important for nutrition  Tiny single-celled zooxanthellae (zooks) that live in coral tissue  Zooks make food for coral and help it deposit skeleton  Coral provides zooks with nutrients How do corals grow  Most reef building corals are colonies of many polyps o Colonies grow as polyps divide o Each polyp builds a “cup” into which it can retreat  Colonies of each coral species have a distinctive shape  Reproduction o Asexual: pieces break off and reattach o Sexual: release eggs and sperm into water- fertilization occurs (SPAWNING OR BROODING)  Some release only sperm and the sperm land in female coral o Larvae that escape predators and settle in a good spot grow into small coral  After many years, corals reproduce and complete their life cycle Chapter 3: Food on Coral Reefs  Acquired by o Predation o Herbivory o Filter feeding o Deposit feeding o Symbiotic microbes  Dependent on type of predator/prey the animal is 1. Swimming and drifting predators  Mobile prey: move around for prey  Characteristics o Stream-lined bodies, large mouths, eyes  Examples: squid, barracuda, shark 2. Sessile prey  Sessile=sitting  Eat prey stuck to bottom  Characteristics o Specifically designed mouths to allow for specialized eating of prey on ground  Examples: hawksbill turtle, Moorish idol fish, butterfly fish, angel fish 3. Bottom-dwelling predators  Stay on ocean floor and use specialized features to eat off ground  Examples: starfish, snails, conch, crabs Defense against predators  Physical features o Shells, toxins, toxic bristles, teeth, claws  Prevent organisms from biting  Examples: snails, slugs, worms, sponges  Defense mechanisms o Camouflage, ink clouds  Predator cant see them  Examples: squid, decorator crab, scorpion fish  Collaborative defense o Schooling  Predator cannot see specific target o Utilize sea anemone o Defensive behavior  Group works together to defend one another strategically  Example: spiny lobster  Physical adaptation to predator  Examples: puffer fish, flying fish Herbivores  Green turtle, parrot fish, sea urchin, snails, surgeon fish, damsel fish  Plants have low levels of nitrogen compared to animals so herbivores are constantly eating to receive adequate nutrients o Grazing  Defense against herbivores o Algae- harder parts o Sponges-toxins o Live in wave-washed areas  Difficult for herbivores to get to the plants because of rough waves o Fast regeneration Sessile filter feeders  Extend filtering structure  Pump water through themselves  Not all filter feeders are small microbes o Whale sharks, manta rays and blue whales are all filter feeders Deposit Feeders  Swallowing dirt/sediment and digesting all organic material Relying on symbiotic microbes  “Manufacture” food through photosynthesis ***Unfair advantage of humans to go against all these genetically designed defense mechanisms depleting species*** Ch.4: Dynamics of Reef Growth and Regeneration Types of Coral Reefs Coastal o Fringing  Most vulnerable to damage by human activities because they are closest to shore Pollutant runoff, pesticides, sediment, sewage o Patch o Barrier  Huge complexes of islands, reefs, lagoon Open ocean o Atoll  Rings of reef around central lagoon Type of reef reflects underlying geology o Darwin found that reefs go from fringebarrier atoll from volcanic activity and growth Reef crest: thick branches oriented into waves o Vigorous growth of corals and coralline red algae “Spur and groove zone” below crest with buttresses of dense live coral Shapes of corals  Suit them for coping with different challenges  Thick branch coral in reef crest  Narrow branch coral (buttresses) o Very vulnerable to breaking  Columns shed sediment and evade competitors  Deep water flat corals suited for deep water o Thin and fragile with algae surface area to collect more sunlight  Tabular corals o Out-compete neighbors for sunlight  Massive forms withstand waves o Don’t grow as fast as thin branched coral because they don’t break easily in hurricanes Growth of coral reef involves Growth processes: o Growth of coral  Coral skeletons are reef framework o Growth of other calcium carbonate  Shell of calcium carbonate (snails, clams, urchins, worms) o Cementation of rubble by growth of coralline red algae  Red/green algaecalcium carbonate  Cement rubble into solid structure Encrusting process: o Binding of rubble by sponges  Sponges stabilize rubble  Areas with less sponges are hard for reef growth Breaking down o Boring sponges and clams make cavities and sediments  Organisms burrow into coral to make home for themselves  Broken off sediment falls into sponged below and turns to rockstability  Grazers generate sediment Parrotfish, sea urchins etc.  Corals colonize CYCLE REPEATS Hurricanes  Normal for coral reefs o Reefs in tropical and subtropical regions o Climate change increased # of hurricanes  Reefs can regenerate after enormous damage  Rebuilding of reefs is same process as building  Coral reef larvae colonize o Reef regeneration requires larvae from healthy reefs in the region Ch. 6: Linked ecosystems- mangroves, sea-grass, meadows & Open Ocean Water flow links:  Mangroves, sea grass meadows, coral reefs, open ocean  Mangroves and sea grass capture and accumulate sediment that flow off the land  Roots retrieve nutrients from sediment o Increases productivity Mangroves  Marine and terrestrial o Tides moving in and out change how much of mangrove is in water  Salty, low-oxygen conditions (in water) are difficult to grown in o Evolutionary tactics have helped mangroves to adapt  Land animals live in mangroves in tops of trees that are always in air  Prop roots host a diverse and dense community o Filter feeders, sponges, oysters, barnacles, anemones, sea squirts, annelid worms, small snails/crabs/worms  Rich hunting ground for predators o Nursery habitat  PREVENT SEA FROM ERODING LAND AND LAND SEDIMENT FROM GETTING INTO OCEAN Sea Grass Meadows  Highly productive o Stabilize sediments, retrieve nutrients from sediments  Are always 100% submerged  Rich hunting ground o Little protection for prey  Only camouflage/armor protect prey o Huge herds of herbivores graze sea-grass  Examples: green turtles, manatees o Fish shelter on reef during day and feed in sea grass at night  Animals move among the reef, grass and mangroves for optimal feeding and protection Ch.7: Complex Webs of Community Interactions Organisms are influenced by physical and chemical factors Each species can simultaneously be prey, consumer, competitor or mutualism Competition  Different organisms require the same limiting resource o Negative relationship for both parties involved (-,-)  Though one organism will win, the energy that is expended makes it a negative interaction  Competition among corals o Blocking sunlight, overgrowing, digesting or stinging neighbors  Herbivory competition o Removal of herbivores allows algae to out-compete corals  Herbivory and pathogen competition o Long-spined black sea urchins  95% killed in 1983 from pathogens  Mobile animals competition o To coexist, similar species must differ by food, habitat or behavior Predation  One organism consumes another o Parasitism or herbivory  Good for one party and bad for other (+, -) Mutualism  Both organisms benefit (+, +) o Clown fish and sea anemone Cascading losses of species  Too much phytoplankton death of sponges o Increase in phytoplankton due to overfishing kills sponges by blocking sunlight o Sponges clear water but are killed by sediment   Sea grasses dying o Sponges absorb algae that harm sea grasses o Sea grasses bind sediment but require sun light  Lobsters die o Lose hiding place  Overwhelming predation  Branching corals grow most rapidly but are most vulnerable to storm destruction Ch. 8: Intimate Associations Symbiosis: individuals of different species living in close association with each other  Commensalism: neither species is harmed, one may benefit (0,0 or +,0)  Parasitism: one species gains at expense of the other (+,-)  Mutualism: both species benefit (+,+)  Cleaning o Cleaner fishes and shrimps pick parasites Microbes  Nutrition (zooks) o Giant clams rely on single celled algae for food o Sea-slug o Many sponges have nitrogen-fixing microbes o Bacteria in sponges may increase growth or manufacture unusual chemicals  Pathogens o Sea fans, corals, sponges Protection from predators Protection from boring organisms o Sponges prevent damage to coral skeletons o Sponges protect mangrove roots from burrowing arthropods Conservation implications of mutualism  Loss of required partner species can doom species to distinction  Mutualisms can become unbalanced if environmental conditions change  At high temperatures zooks abandoned corals “bleaching” Ch. 9: Humans and Coral Reefs  Coral reefs can regenerate after disturbances if they are infrequent and if larvae come from healthy reefs  Coral reefs are overwhelmed when disturbances are too big, too continuous, too novel (oil spills), too numerous o Climate changeincreased hurricanes  Every time forests are cut down, sediment from land washes into reefs o Mangroves can capture this sediment but are being cut down  Excess nutrients o Fertilizers  Fuel algae  Drown corals  Oil smothers corals and mangroves o Moves on currents, floats, combines into tar blocks  Overfishing increased algae (nothing eating it) coral problems  Spawning aggregations  Destructive fishing o Explosives o Shrimp farms  Tourism o Development, diving  Climate change  Diseases What can we do Regional cooperation o Larvae, pathogens, oil travel on currents o Conserve mangroves and sea grasses Tourism solutions o Diver training Mooring buoys Marine protected areas Respect fishing Grow resistant coral

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Chapter 16, Problem 48 is Solved
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Textbook: Calculus: Early Transcendental Functions
Edition: 6
Author: Ron Larson
ISBN: 9781285774770

Calculus: Early Transcendental Functions was written by and is associated to the ISBN: 9781285774770. Since the solution to 48 from 16 chapter was answered, more than 262 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Calculus: Early Transcendental Functions, edition: 6. The answer to “In Exercises 47 and 48, use a Taylor series to find the first terms of the series solution of the differential equation under the specified initial conditions. Use this polynomial to approximate for the given value of x.” is broken down into a number of easy to follow steps, and 38 words. This full solution covers the following key subjects: use, Series, polynomial, equation, exercises. This expansive textbook survival guide covers 134 chapters, and 10738 solutions. The full step-by-step solution to problem: 48 from chapter: 16 was answered by , our top Calculus solution expert on 11/14/17, 10:53PM.

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Solved: In Exercises 47 and 48, use a Taylor series to