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5–10, compute the Fourier sine series for the given

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider ISBN: 9780321747730 43

Solution for problem 5E Chapter 10.4

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Fundamentals of Differential Equations | 8th Edition

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Problem 5E

In 5–10, compute the Fourier sine series for the given function.

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Notes from 3/15-3/17 I. II. Mechanism of Contraction  In the absence of Ca+- tropomyosin blocks the interaction of thick & thin filaments  In the presence of Ca+- calcium will bind TNC (subunit of troponin which binds calcium); this will cause the tropomyosin block to be removed 1. Crossbridge (Myosin Head)- now will bind actin; ADP + Pi (phosphate) bound to myosin head 2. Power stroke- (where contraction actually takes place) the crossbridge changes shape; ADP + Pi (phosphate) are released 3. Crossbridge detachment- myosin head separates from thin filament; requires binding of ATP 4. Cocking step- crossbridge returns to original conformation/structure; ATP --> ADP + Pi 5. Can restart cycle- must have ATP and Calcium  Rigor mortis- the stiffness in muscle several hours after death caused by: 1. Calcium present in sarcoplasm so tropomyosin block is removed, can no longer pump Ca+ back into sarcoplasmic reticulum 2. No ATP, no metabolism so crossbridge detachment cannot take place so myosin head is stuck to actin filaments III. Regulation of Contraction  Skeletal muscle is stimulated by nerves  Regulation of calcium  Neuromuscular junction- junction of a nerve and a muscle (pictures on Slide 17)  Axon terminal- the very end of a nerve  Synaptic vesicles found in axon terminal- in nerves that supply skeletal muscles there is the neurotransmitter acetylcholine  Synaptic cleft- the physical space between the axon terminal and muscle  Synapse- the meeting/joining of a muscle and a nerve or a nerve and a nerve  Action potential 0. Goes down the nerve 1. Will stimulate the fusion of synaptic vesicles with nerve membrane, releasing (ACH) acetylcholine into the synaptic cleft 2. ACH will diffuse across the cleft & bind to receptors on muscle fiber sarcolemma 3. This signal (action potential) will spread out over the sarcolemma 4. T-tubules cause sarcoplasmic reticulum to release Ca+---> mechanism of contraction takes place IV. Reset 1. Enzyme- acetylcholine esterase is found in the synaptic cleft, which destroys acetylcholine about as fast at is made, so ACH is present in synaptic cleft for only a short amount of time 2. ACH can be taken up my axon terminal cells- endocytosis 3. ACH can diffuse away 4. ATP pump- Ca+ is pumped back into the sarcoplasmic reticulum V. Twitch Contraction  Myogram- graph 1. Latent Period- between when muscle stimulated to contraction 2. Period of Contraction 3. Period of Relaxation  Muscles differ in periods of contraction of relaxation  Picture of above depicts a single twitch V. Smooth Contractions 1. Increase in frequency of stimulation  Tetanus- very high frequency of stimulation; stimulate muscle very fast; no relaxation at al between stimuli 2. Multiple Motor Unit Summation  Motor unit- consists of a nerve + the fibers it stimulates o on avg. there around 4-200 muscle fibers in a motor unit o there is more than 1 motor unit per muscle o Fibers of a motor unit are not clustered together 3/17/16  Motor unit A and Motor unit B alternate to create constant tension (smooth)  Isometric contraction- load > tension  Isotonic contraction- tension > load  Most contractions are a combo of the above I. Factors that Affect Force (Tension) of a Contraction 1. Warm up - treppe (SLIDE 29) 2. Size of Muscle 3. Number of Motor Units that are contracting 4. Muscle stretch  Max force of a muscle is generated when the most myosin heads are in contact/binding with actin (SLIDE 36) II. Energy  ATP ---> ADP + P ---> energy  Soluble ATP in the cytoplasm, which will last about 6 seconds  Creatine-P (phosphate)- molecule found in sarcoplasm (Creatine-P + ADP --> creatine + ATP, which whill last about 10-15 seconds); creatine-P + ADP = creatine phosphokinase d. Aerobic respiration: glucose + O2 --> CO2 + H2O + energy; glucose is broken down step by step, the energy is captured to form ADP + Pi --> ATP; glycogen can be broken down to glucose, various fatty acids can also fit into this, but usually during exercise, glucose is the main one i. Aerobic Pathway- when there is plenty of O2 (SLIDE 41) 1. Glycolysis (takes place in cytoplasm): glucose (br) (some ATP) --> 2-pyruvic acid (3 carbons) 2. Oxidative phosphorylation mitochondrion: 2-pyruvice acid--> mitochondria--> CO2, O2, ATP with about 20 times more ATP than glycolysis ii. Anaerobic Pathway- no O2, enlarge and engorge muscles, pinching of blood vessels (SLIDE 40) 1. Glycolysis (takes place in cytoplasm): glucose (br) (some ATP) --> 2-pyruvic acid (3 carbons) 2. Pyruvic Avid --> lactic acid, diffusing into blood stream, some in liver, some stays in muscles, doesn't go to mitochondria; feel legs burn (accumulation of lactic acid), can do this for a little while, not forever, ATP is very quickly created iii. Summary 1. Short exercise: a. to about 15 seconds  Soluble ATP, creatine-P b. Exercise to about 60 seconds  Anaerobic respiration: glycolysis, OPM 2. Long: a. Hours  Aerobic respiration  Oxygen debt- more ATP being used than being generated by aerobic pathway e. Muscle Fatigue  ATP production less than usage  Cramps- stiffness in muscle, caused by myosin head remaining bound to actin thin filaments, in order for it to release, ATP must bind o Accumulation of lactic acid- acidify muscle, muscle will not function properly in an acidic environment o Loss of Na+ and K+- sodium and potassium, salts necessary for conduction of action potentials III. Fiber Types (SLIDE 45) . Slow Oxidative Fibers- fatigue resistant  Slow- slow myosin, not going to contract very fast, long period of contraction  Oxidative- specialized for aerobic pathway- lots of mitochondria  Myoglobin- will bind O2, storing O2, red color fibers  Low amounts of glycogen, run on glucose which is either present or brought to the cells via bloodstream a. Fast Glycolytic Fibers- sprinters have these  Fast- fast myosin, muscles are going to contract  Large amounts of glycolytic enzymes, not so many mitochondria  Broad fibers  Fatigueable  Relatively high levels of glycogen  No myoglobin- white color fibers b. Fast Oxidative Fibers- intermediate  Relatively fast myosin- 10 or more genes that code for different myosin's, so each fiber type has a different myosin  Oxidative  Specialized for aerobic respiration- lots of mitochondria  Myoglobin- intermediate amount --> pink color fibers  Some glycogen- low levels  Most muscles are going to be a mixture  All the muscle fibers of a different motor unit are the same type (FG, SO, or FO) IV. SMOOTH MUSCLE  Not as highly ordered as skeletal muscles with striations  Contains thick and thin filaments (1:13 ratio), (1:3 ratio in skeletal)  No troponin, no t-tubules  No sarcomeres, no myofibrils  But there are bundles of filaments (similar to myofibrils)  Thin filaments attach dense bodies of surface of cells, intermediate filaments also attach to these dense bodies  Caveolae- pocket on cell surface, high in Ca+  Lack elaborate coverings  Arranged in sheets- 2 layers  Circular- wrap around tube  Longitudinal- parallel to axis of tube  Surround digestive system, tubes of respiratory system, etc.  Lack highly structured neuromuscular junction  Very wide synaptic cleft- diffuse junctions  Varicosities- come in contact with sheet of smooth muscle, have synaptic vesicles, release neurotransmitters

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Chapter 10.4, Problem 5E is Solved
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Textbook: Fundamentals of Differential Equations
Edition: 8
Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider
ISBN: 9780321747730

Since the solution to 5E from 10.4 chapter was answered, more than 271 students have viewed the full step-by-step answer. The answer to “In 5–10, compute the Fourier sine series for the given function.” is broken down into a number of easy to follow steps, and 11 words. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations , edition: 8. This full solution covers the following key subjects: compute, fourier, function, given, Series. This expansive textbook survival guide covers 67 chapters, and 2118 solutions. Fundamentals of Differential Equations was written by and is associated to the ISBN: 9780321747730. The full step-by-step solution to problem: 5E from chapter: 10.4 was answered by , our top Calculus solution expert on 07/11/17, 04:37AM.

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5–10, compute the Fourier sine series for the given