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Solved: Find f (2), f (3), f (4), and f (5) if f is

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 4E Chapter 5.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 4E

Problem 4E

Find f (2), f (3), f (4), and f (5) if f is defined recursively by f (0) = f (l) = 1 and for n = 1.2,…

a)f ( n + 1) = f (n) - f (n − 1).

b) f ( n + 1) = f (n) f (n − 1).

c) f ( n + 1) = f (n) 2 + f (n − l)3.

d) f ( n + 1) = f ( n) f ( n − 1)

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Food poisoning  Food-borne illness: illness caused by food  Usually causes gastrointestinal symptoms  Abdominal pain, nausea, diarrhea, and vomiting  Can cause kidney failure, arthritis, paralysis, miscarriage, death  Usually caused by microbes (microorganisms), such as bacteria, viruses, fungi, or parasites  Microbes that can cause disease are pathogens (they generate pathology) NOTE: only2 cases of food poisoning that can be tied back to the same source are needed before it can be considered an outbreak Food-borne illness: infection vs. intoxication  Food-borne infection: caused by pathogens that multiply in the human body  Usually from consumption of a large number of pathogens that cause infection or produce toxins in the body  Example: Salmonella  Food-borne intoxication: caused by consuming food containing toxins produced by pathogens  Can be caused byfood containing onlya few pathogens if theyhave produced enough toxin  Example staphylococcus aureus Bacterial Food-Borne illness Food poisoning symptoms and severity depends on:  Potency of contaminant  How much of is consumed  How often it is consumed  Age, size, nutritional status, chronic diseases  Absorption, metabolism, storage in the body  Immune function  (at risk: young, elderly, pregnant women, people with AIDS or on chemotherapy or immunosuppressant drugs) Federal Agencies  Set standards and establish regulations for:  safe handling of food and water  information included on food labels  Regulate use of additives, packaging materials, and agricultural chemicals  Inspect food processing and storage facilities  Monitor domestic and imported foods  Investigate outbreaks of food-borne illness Food Supply Monitoring Agencies Recent Federal Changes National Food Safety Initiative  Goal: reduce food-borne illness by improving US food safety practices and policies  Targets food safety from farm to table FDA Food Safety Modernization Act  Focus: preventing food-borne illness  Passed in 2011 in response to the continued threat from our food supply  FDA: inspection mandate and new legal powers Hazard Analysis Critical Control Point (HACCP)  Food safety system required for food manufacturers, processors, and distributors  Analyzes food production, processing, and transport  Goal: identify potential sources of contamination and points where measures can be taken to prevent contamination  Monitors these critical control points Bacteria  In soil, on our skin, on most surfaces in our homes, and in food  Most are harmless, some are beneficial, and a few are pathogenic Examples of Bad Bacteria Salmonella  Most common cause of US food-borne illness  Common sources: Poultry and eggs are the foods most commonly contaminated;  Prevention: killed by heat; thoroughly cook foods likely to be contaminated E coli  Causes: some strains are harmless; others can cause serious food-borne infection  E. coli O157:H7 produces a toxin causing abdominal pain, bloody diarrhea, fatal kidney failure  Common sources: fecal contamination of water or food (for example, meats and produce) Listeria Monocytogenes  Causes: flulike symptoms; more serious in high-risk groups (pregnant women, children, elderly people, with compromised immunity)  During pregnancy: causes spontaneous abortion and stillbirth, fetal meningitis and blood infections  Common sources: everywhere in environment  Prevention: survives and grows at refrigerator temperatures so infects ready-to-eat foods; heat hot dogs and lunchmeats to steaming point and avoid unpasteurized dairy products NOTE: listeria fatality rate is 21% How Bacteria Contaminate Food  Raw & unpasteurized foods  Close/Dirty living environments: bacteria & bacteria-related illness is spread through livestock population  Uncooked/Unwashed foods: bacteria on the surface are not killed or removed  Contamination in factory line: machine parts get covered in bacteria & contaminate all food that passes through it  Sick employees: sick food handlers cough or sneeze on food, spreading their germs  Ground meat products: bacteria mixed throughout meat in grinding process & may not be killed by cooking if not cooked thoroughlEX: hamburger w/ pink in middle Botulism Clostridium Botulinum bacteria  Causes: blocked nerve function, resulting in vomiting, abdominal pain, double vision, and paralysis leading to respiratory failure and death; deadliest of all bacterial food toxins  Common sources: in soil, water, and animal intestinal tracts; toxin produced when heat-resistant spores grow in low-oxygen, low-acid conditions; found in improperly canned foods and foods held in large containers NOTE: this is the deadliest of bacteria Infant botulism  Most common form of botulism in the US  Caused by ingestion of botulism spores (most common is honey) Spores germinate in the infant’s gastrointestinal tract, producing toxin  In adults, competing intestinal microflora prevent spores from germinating  Some toxin is absorbed into the bloodstream, causing weakness, paralysis, and respiratory problems; infants generally recover  Never feed honey to children less than 1 year Viruses  Not classified as living or as cells since they cannot reproduce on their own  Human viruses reproduce only inside human cells  Viruses turn human cells into virus-producing factories  Viruses that cause human diseases cannot grow and reproduce in foods Noroviruses  Group of viruses  Cause: about 50% of all US food-borne gastroenteritis (stomach inflammation), A.K.A. “stomach flu”  Common sources: eating food contaminated with virus or touching contaminated surface and then putting fingers in mouth; Shellfish can be contaminated in water polluted with feces  Prevention: cooking destroys noroviruses NOTE: noroviruses are the #1 source of food outbreaks, as they can live on surfaces for up to 2 weeks, and some ordinary cleaners don’t kill it Mold & Fungus  Many types grow on foods such as bread, cheese, and fruit  Under certain conditions, molds produce toxins (>250 different mold toxins)  Cooking and freezing stop mold growth but do not destroy toxins already produced  If food is moldy, discard it, clean the area where it was stored, and check neighboring foods to see if they are contaminated Mold & Liver Cancer Parasites  Organisms that live at the expense of others  Some are microscopic single-celled animals; others are worms large enough to be seen with the naked eye  Prevention: killed by thorough cooking  Giardia lamblia, Cryptosporidum parvum, Trichinella spiralis (Multicellular parasite in raw or undercooked pork Pathogens in Food: Prions  Pathogenic protein that causes degenerative brain diseases such as spongiform encephalopathies  Short for proteinaceous infectious particle Prion diseases  Bovine spongiform encephalopathy (BSE): deadly neurodegenerative disease in cattle  Creutzfeldt-Jakob Disease (vCJD): human BSE  Causes: eating brain, nervous tissue, intestines, eyes, or tonsils from cow infected with BSE (meat and milk have not been found to transmit prions)  Symptoms: mood swings and numbness progressing to dementia and death  Prevention: prevent US cattle from contracting BSE (cooking does not destroy prions) Prevent microbial food-borne illnesses  Choose food carefully; when in doubt, throw out  Prepare food in a clean kitchen to reduce cross-contamination (transfer to another food) NOTE: cross-contamination happens most commonly in home kitchens  Store food in refrigerator or freezer  Foods served cold should be kept cold until served  Thaw frozen foods in refrigerator or microwave (not at room temperature)  Heat foods to recommended temperatures  Cooked foods should be kept hot until served Safe Grocery Decisions Leftovers  Minimize the danger zone  Chill quickly (within 2 hours or less)  Store in small, shallow containers  Wrap well  Good in refrigerator 3 to 4 days  Reheat to 165 Safe Handling, Storage, & Prep clean separate cook chill Contaminants in the Food Supply Bioaccumulation Pesticides in Food  Prevent plant diseases and insect infestations  Applied both before and after harvest  Produce higher yields and look more appealing from less insect damage  Found on treated plants and in meat, poultry, fish, and dairy products since they enter water, soil, and other parts of environment  Risks depend on the size, age, and health of the consumer and on the type and amount consumed Pesticides regulation  Types of pesticides, how often used, and amount of residue that may remain when foods reach consumers are regulated  EPA: approves and registers pesticides used in food production and establishes tolerances (maximum amounts of pesticide residues that may remain in or on a food)  FDA and USDA: monitor pesticide residues in foods Organic Foods  Produced, processed, and handled according to USDA National Organic Program standards  Reduced chemical pesticides and fertilizer use  USDA determines substances that can or cannot be used  USDA must certify before labeled “Organic”  Most conventional pesticides, synthetic fertilizers, sewage sludge, genetically modified ingredients, irradiation, antibiotics, and growth hormones are prohibited  Recycling of resources  Conservation of soil and water Labeling Organic Foods Industrial Contaminants Polychlorinated Biphenyls (PCBs)  Group of carcinogenic compounds  Prior to 1970s, PCBs in runoff from manufacturing plants’ contaminated water  Still in environment and accumulate in fish from contaminated waters since PCBs do not degrade  Prenatal exposure and contaminated breast milk causes nervous system damage and learning deficits  Check with local health departments for recommendations regarding fish consumption during pregnancy and lactation Other Manufacturing Contaminants  Chlordane (used to control termites)  Radioactive substances  Cadmium: interferes with mineral absorption, causes kidney damage, impairs brain development  Lead  Arsenic: increases cancer risk  Mercury: damages nerve cells; more damaging during prenatal development  Bisphenol A (BPA) Antibiotics in Animals  Animals are treated with antibiotics when sick or to prevent disease and promote growth  Increases meat production and reduces costs  If used improperly, antibiotic residues can remain in meat  Creates antibiotic-resistant bacteria Hormones in Animals  Used to increase weight gain in sheep and cattle and milk production in dairy cows  Some hormones, such as estrogen and testosterone, occur naturally  Generally administered in slow-release form, and levels are no higher in treated animals than in untreated animals  Must demonstrate that synthetic hormone residues in meat are within safe limits Bovine Somatotropin (bST)  Has created public concern  Produced naturally by cows and stimulates milk production  Genetically-engineered synthetic hormone is produced by bacteria and injected into cows to further increase milk production  Milk from cows that have been treated with genetically-engineered bST is indistinguishable from other milk Minimize food contaminants  Choose a wide variety of foods  Choose organic or locally-grown produce  Wash and in some cases peel produce  Trim fat from meat and remove poultry skin  Choose wisely and consume a variety of fish  Remove fish skin, fatty material, and dark meat  Broil, poach, boil, and bake fish Factors to Keep Food Safe Food Preservation  Older methods: heating, cooling, drying, smoking, adding substances (sugar or salt)  Newer methods: irradiation, specialized packaging  Risk: substances enter food High & Low Temp Preservation  Provide appealing, safe foods  Cooking: kills microbes, destroys most toxins  Pasteurization: process of heating food products to kill microbes  Sterilization and aseptic processing: placement of sterilized food in sterilized package using sterile process  Refrigeration or freezing does not kill microbes but slows or stops microbial growth Food Irradiation (cold pasteurization)  Used in more than 40 countries and used infrequently in the US because of public suspicion and of lack of irradiation facilities  Exposes food to high doses of X-rays, gamma radiation, or high-energy electrons to kill microbes and insects and inactivate enzymes that cause germination and ripening of fruits and vegetables  Food additive: produces compounds not present in the original foods (regulated) Modified atmosphere packaging (MAP)  Preservation technique to prolong shelf life of processed or fresh food by changing the gases surrounding food in the package  Uses packaging materials impermeable to gases  Air in package is vacuumed out to remove oxygen  Product remains in a vacuum or the package is infused with another gas  Lack of oxygen prevents aerobic bacteria growth, slows ripening and oxidation reactions Additives vs. Contaminants  Food additive: any substance that can be expected to become part of a food  FDA regulates the types and amounts of food additives  Accidental contaminants: unexpected substances that enter food  Not regulated Regulating Food Additives  Prior-sanctioned substances: > 600 additives in use when legislation was passed  Generally recognized as safe (GRAS): additives considered safe based on history of use in food before 1958 or on published scientific evidence  Can be added only at levels 100 times below highest level shown to have no harmful effects  Part of the 1958 Food Additive Amendment states that a substance that induces cancer in either an animal species or humans, at any dosage, may not be added to food  If new evidence suggesting additive in either category is unsafe, FDA may require removal Sensitivities to Additives  Some individuals are allergic or sensitive to certain food additives  Monosodium glutamate (MSG) can cause MSG symptom complex or Chinese restaurant syndrome  Sulfites can cause symptoms ranging from stomach ache and hives to severe asthma  FD&C yellow no. 5 (listed as tartrazine on medicine labels) may cause itching and hives Carcinogens from cooking meats  Polycyclic aromatic hydrocarbons (PAHs): formed when fat drips on a grill and burns then rise with smoke and deposited on food surface  use lower-fat meat and a layer of aluminum foil to prevent fat from dripping on coals  Heterocyclic amines (HCAs): produced by burning of substances in meats  precook meat, marinate meat before cooking, cook at lower temperatures, reduce cooking time with smaller pieces of meat, avoid overcooking Genetically modified (GM) crops  Gene (piece of DNA) for desired characteristic (for example disease resistance) is transferred from plant, animal, or bacterial cells into plant cells  Creates recombinant DNA - a combination of DNA from two organisms  Modified cells divide and differentiate into plant  New plant is a transgenic organism  Each cell in plant contains transferred gene  Most common: soybeans, corn, rapeseed (canola) Growth of GM Crops Biotechnology for malnutrition  For protein deficiency: corn, soybean, and sweet potato varieties with enhanced essential amino acids levels  For vitamin A deficiency: genes for carotene synthesis enzymes inserted into rice  To address multiple nutrient deficiencies: cassava with increased zinc, iron, protein, and vitamin A levels Biotechnology Concerns  Nutrient content may be negatively affected  Allergen or toxin may be introduced  GM crops will be used to the exclusion of other varieties, reducing biodiversity which may reduce ability to adapt to new conditions, diseases, or other hazards  GM crops may create “superweeds”  GM crops producing pesticides may promote evolution of pesticide-resistant insects Labeling of GM Foods  Not required to have special labeling unless:  nutritional composition has been altered  it contains potentially harmful allergens, toxins, pesticides, or herbicides, or new ingredients  it has been changed significantly enough so that its traditional name no longer applies  Premarket approval required if new food contains substance not commonly found in foods or without a history of safe use in foods

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Chapter 5.3, Problem 4E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

Since the solution to 4E from 5.3 chapter was answered, more than 318 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 4E from chapter: 5.3 was answered by , our top Math solution expert on 06/21/17, 07:45AM. This full solution covers the following key subjects: defined, Find, recursively. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. The answer to “Find f (2), f (3), f (4), and f (5) if f is defined recursively by f (0) = f (l) = 1 and for n = 1.2,…a)f ( n + 1) = f (n) - f (n ? 1).b) f ( n + 1) = f (n) f (n ? 1).c) f ( n + 1) = f (n) 2 + f (n ? l)3.d) f ( n + 1) = f ( n) f ( n ? 1)” is broken down into a number of easy to follow steps, and 80 words. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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