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Answer: Use a graphing device to find all real solutions

Precalculus: Mathematics for Calculus | 6th Edition | ISBN: 9780840068071 | Authors: James Stewart ISBN: 9780840068071 161

Solution for problem 73 Chapter 3

Precalculus: Mathematics for Calculus | 6th Edition

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Precalculus: Mathematics for Calculus | 6th Edition | ISBN: 9780840068071 | Authors: James Stewart

Precalculus: Mathematics for Calculus | 6th Edition

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Problem 73

Use a graphing device to find all real solutions of the equation.x 4 3x 3 3x 2 9x 2 074.

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Shira Clements BSCI105 Chapter 8- Introduction to Metabolism Cellular Respiration- drives cellular economy by getting the energy stored in sugars and other fuels Metabolism- totality of organism’s chemical reactions - Arises from orderly interactions between molecules - Manages the material and energy of cell o Transforms matter and energy Organization of the Chemistry of Life into Metabolic Pathways - Metabolism is like a road map of chemical ractions in a cell, with metabolic pathways- begins with a specific molecules and then it is changed by many steps which then results in a new product. o Each step of the way, the molecule is catalyzed by a specific enzyme o Mechanisms that regulate enzymes balance metabolic supply and demand (like traffic lights) - Catabolic Pathways- break down complex molecules and make simpler ones, so by doing that energy is released (i.e. cellular respiration) - Anabolic Pathways (also known as bio synthetic)- build complex molecules from simple ones, so they need to consume energy (i.e making proteins) o the energy released from catabolic can be used for anabolic (downhill and uphill) Energy- capacity to cause change, ability to rearrange matter o Work- move matter against opposing forces o Life depends on it- cells use it to transform energy from one to another o Kinetic Energy- energy associated with motion o Heat/ Thermal Energy- kinetic energy associated with the random movement of molecules o Potential Energy- energy that matter possess because of location or structure o Chemical Energy- refers to potential energy available for release in a chemical reaction  Biochemical pathways (in cellular structures) allow us to release chemical energy from food molecules ad use the energy to power life processes  Movement can change certain energy- i.e from kinetic to potential because of height Thermodynamics- study of energy transformations that occur in matter - Open system- energy and matter can be transferred between the system and surroundings- organisms - Isolated system- energy and matter cannot be exchanged between system and surroundings - First law of thermodynamics- energy can be transferred or transformed, but it cannot be created or destroyed (some becomes unavailable to do work)- principle of conservation of energy - System can only put heat to work when there is a temperature difference that results in heat flowing from a warmer location to a cooler one. If temperature is uniform, then only use for heat energy is to make body warmer (like in humans) - Second law of thermodynamics- every energy transfer or transformation increases the entropy of the universe- unstoppable trend to randomness. For a process to occur spontaneously, the must increase the entropy of the universe. o Entropy- measure of disorder/randomness, causes much of the loss of usable energy during an energy transfer.  Living systems increase entropy of surroundings- take organized form of energy and replaces it with less ordered forms- when breaking down complex molecules, it releases carbon dioxide and water, which are smaller molecules that possess less chemical energy than the food and then accounted by heat during metabolism.  Spontaneous Process- process that occurs without input of energy (spontaneous means energetically favorable.  Nonspontaneous Process- process that will only happy if energy is added to the system.  Water moves downhill spontaneously, but uphill with an input of energy. Free Energy Change G - Free energy- portion of system’s energy that can perform work when temperature and pressure are uniform throughout the system- during chemical reaction mostly. o G=H-TS o H= change in system’s enthalpy- total energy in system- when negative, that means it gives up enthalpy, so H decreases o S= change in system’s entropy o T= absolute pressure in Kelvin, which is 273+Celcius o Once we know G- we can tell if the process will be spontaneous, which will only happen when G is negative. Either H or TS must be negative, or both, in order to be spontaneous o G= G final stainitial stateuse less free energy, the system in final state is less likely to change and therefore more stable than it was previously. o Can think of it as measure of instability (tendency to change to more stable state)- unstable= higher G and more stable=lower G o Equilibrium- maximum stability.  Chemical equilibrium is when forward and backward reaction occur at same rate, and no net charges in concentration of reactants and products  When it goes toward equilibrium, the free energy of reactants and products decrease- free energy increases when it is pushed away from equilibrium.  Any change from equilibrium G will be positive and will not be spontaneous because work will have to be input- a process is spontaneous and can perform work only when it is moving toward equilibrium Free Energy Applied to Metabolism - Exergonic Reaction- energy outward- net release of energy o G is negative because it loses free energy o occur spontaneously o magnitude of G for an exergonic reaction represents the maximum amount of work the reaction can perform- greater decrease of free energy, the more work that can be done o cellular respiration- turning glucose and O i2to water and CO 2 - Endergonic Reaction- one that absorbs free energy from surroundings o Essentially stores free energy in molecules (G increases), G is positive o Nonspontaneous o G represents amount of energy needed for reaction to occur Equilibrium and Metabolism - at equilibrium in an isolated system, there is no work. Chemical reactions of metabolism are reversible and would reach equilibrium if they were in isolated system. o A cell that reaches metabolic equilibrium is dead because they can’t do work o Constant flow in and out, so metabolic pathway never reach equilibrium, because it is an open system. - Some reversible reactions are pulled in one direction, making it impossible to reach equilibrium- product does not accumulate, rather it just becomes reactant in next step (it is steps and used in next reaction) o Occurs because there is a huge free energy difference between glucose and oxygen at top of energy hill and bottom of energy hill- cells have to have a constant supply of glucose and O an2 expel waste products in order for metabolic pathways never reach equilibrium and continue with work Cell Does Three Kinds of Work- - Chemical work- endergonic reactions - Transport work- pumping substance across membrane against direction of spontaneous movement - Mechanical work- contraction of muscle cells, cilia beating… Energy Coupling- - Manner in which cell manages energy resources to do work - Use of exergonic process to drive endergonic process o ATP is responsible for mediating and many times it is source of energy that powers cellular work - ATP o Contains ribose sugar, adenine as nitrogenous base, and 3 phosphate groups- bond between phosphate can be broken by hydrolysis, which it then becomes ADP, which is exergonic reaction and releases 7.3 kcal per mole of ATP hydrolyzed. G=- 7.3- under standard conditions, but cells are not in standard conditions, so G in cells is around -13 kcal/mol. o Release of energy in hydrolysis comes from chemical change to a state of lower free energy, not from the breaking of phosphate bond- it releases more energy than most other molecules o When hydrolyzed, release of energy releases heat and cells intake it to perform their work, which helps with endergonic reactions.  If G of an endergonic reaction is less than the amount of energy released by ATP (exergonic), then they can be coupled to be a complete exergonic reaction-this usually involves transfer of phosphate from ATP to another molecule (possibly the reactant)- phosphorylated intermediate- one that gets the phosphate, which is usually more reactive/less stable.  ATP can lead to change in protein shape and can bind with other molecules (phosphorylated intermediate helps) o We use ATP all the time, and just by adding a P to ADP, it is regenerated- the free energy required to phosphorylate ADP comes from the exergonic breakdown reactions (catabolism) in cell.  ATP Cycle- couples cell’s exergonic to exergonic reactions  Works really fast  Regeneration of ATP from ADP and P is endergonic  ATP hydrolysis to ADP and P yields energy which is used for work, then energy from catabolic pathways is used and ADP and P are now ATP. Enzymes - Macromolecule that acts as a catalyst- chemical agent that speeds up a reaction without being consumed by reaction - Has specific locations within cell- has compartments - Reactant molecules must absorb energy to reach the state where bonds can change- once new bonds of product molecules form, energy is released and molecules return to stable shapes with lower energy than contorted state- have to change starting molecule to unstable state for reaction to begin. - Activation Energy E - enArgy required to contort reactant molecules so bonds can break, or initial investment of energy for starting reaction. Amount of energy needed to push reactants to top of hill in order to be able to go down (their goal). o Can be in form of heat- molecules collide more often and making bonds break more easily, but not in organisms, because it can be dangerous since it speeds up all reactions and kills many others - Transition state- when molecules have absorbed enough energy for bonds to break and reactants are in unstable condition- they are activated already. o After, when they are rearranging to new bonds, more energy is being released than E hAd. - Enzyme is used in organisms and lower the E barrAer- allows reactant to absorb less energy to reach the transition state o Will not change G for reaction o Makes it possible for cell to have a dynamic metabolism o Specific for reactions they catalyze, so determine what will happen when - Substrate- reactant enzyme acts one- enzyme binds to substrate at active site (pocket or something similar on face of enzyme) forming enzyme-substrate complex- held there by weak interactions. Then once this happens, enzyme converts reactant to product. o very specific because of the amino acid sequence it is made of- has a compatible fit for substrate o enzyme is not set to one shape in equilibrium with slight differences in free energy for each position o Induced fit- as substrate enters active site, enzyme changes shape because chemical changes occur when it binds to it- bring chemical groups of active site into position that enhance ability to catalyze chemical reaction- interactions of side chains. Enzyme hugs the substrate more this way. o Very small amount of enzyme can have a huge impact in catalytic cycle o Can catalyze either reverse or forward reaction- depending on which direction has negative G. - Mechanisms that lower E - aAtive site provides template on which substrates come together in proper orientation o Active site of enzyme clutches the bound substrate- may stretch molecule toward transition state form so it goes quicker, which stresses and breaks the critical chemical bonds that must break during reaction. o Active site can provide a microenvironment that is favorable to that reaction o Active site can provide covalent bonding between substrate and side chain of enzyme. o The more substrate molecules that are available, the more frequently they access the active sites of the enzyme  At some point, the concentration of substrate will be high enough that all enzyme molecules have their active sites engaged. When this happens, the enzyme is called saturated, and the rate is determined by speed at which the active site converts substrate to product. Only way to increase rate then is to add more enzyme. - Activity of enzyme is effected by environment- pH, chemicals, and temperature. o Optimal conditions- conditions under which the specific enzyme works best in o Up to a point- rate of enzyme increases with increasing temperature- because substrates collide with enzyme more, but above that temp, the speed of the enzymatic reaction drops significantly because the thermal energy would disrupt all the bonds that stabilize the stability of the enzyme (then denature)  Most human enzymes have optimal temp of 35-40 C  Optimal pH of 6-8 in humans usually- pepsin in stomach has pH of 2 - Cofactors- non protein that helps the enzyme catalyze o Can be attached to enzyme o If organic- it is known as coenzyme- vitamins provide this - Enzyme inhibitors- certain chemicals that stop the enzyme from working properly o if it covalently bonds to the enzyme, it is irreversible o some bond to enzyme with weak interactions, it is reversible  Competitive Inhibitors- mimic the normal substrate and compete to bind to active site, so it reduces productivity. Can be overcome by increasing concentration of substrate, so it would have a better chance of entering the active site.  Noncompetitive Inhibitors- bind to another part of enzyme, which changes enzymes shape, so active site is less effective. - Regulation of Enzyme Helps Control Metabolism o Allosteric Regulation- when a protein’s function at one site is affected by binding of a regulatory molecule to a separate site, which can cause inhibition or stimulation- can work like a noncompetitive inhibitor  Chaos would occur if all metabolic pathways worked at the same time, so the enzymes are there to regulate and control it.  Constructed from two or more subunits usually (polypeptide chain and own active site) that switch between an active and inactive form  Simplest- activating molecule binds to a regulatory/allosteric site, which stabilizes shape of functional active sites, when the inhibitor stabilizes inactive form of enzyme. When subunit binds and changes the shape of one active site, all the other active site’s shape change to that shape.  EX- ATP binds to catabolic enzymes- lowering affinity for substrate which inhibits activity. However, ADP is an activator of same enzymes- logical because catabolism generates ATP.  Cooperativity- substrate molecule binds to multi-subunit enzyme which trigger shape change in all subunits, which increases activity at other active sites. (ex- hemoglobin, even though it is not an enzyme)  Increases response of enzymes to substrate- one substrate primes an enzyme to act on more substrate molecules more readily.  Hard to isolate because bind to enzyme at low affinity  Very high specificity o Feedback inhibition- metabolic pathway is switched off by inhibitory binding of end product to an enzyme that acts early in pathway (ATP allosterically inhibits an enzyme in an ATP generating pathway) and it saves the cell from producing unnecessary chemicals

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Chapter 3, Problem 73 is Solved
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Textbook: Precalculus: Mathematics for Calculus
Edition: 6
Author: James Stewart
ISBN: 9780840068071

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Answer: Use a graphing device to find all real solutions