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Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This

Linear Algebra with Applications | 4th Edition | ISBN: 9780136009269 | Authors: Otto Bretscher ISBN: 9780136009269 434

Solution for problem 32 Chapter 1.2

Linear Algebra with Applications | 4th Edition

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Linear Algebra with Applications | 4th Edition | ISBN: 9780136009269 | Authors: Otto Bretscher

Linear Algebra with Applications | 4th Edition

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Problem 32

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points (a, , /?,) are given to you, and your job is to connect the dots in a reasonably smooth way. Let aj+ \ > ai.One method often employed in such design problems is the technique of cubic splines. We choose ft (r), a polynomial of degree < 3, to define the shape of the ride between (a, _ i, _ 1) and (a,-,/?/), for / = l,,..,/j.Obviously, it is required that ft (at) = bi and ft (a/_ i) = bj - 1, for i = 1,..., n. To guarantee a smooth ride at the points (ai,bi), we want the first and the second derivatives of ft and //+ 1 to agree at these points:= //+](,) fliat) = f!'+x(a;).and for / = 1,. ,n - 1.Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required thatf\ (00) = /> i.) = 0.Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)

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UMassAmherst Management260 ProfessorMalkovich Week12Notes:Apr11–15 ByEunjiCho Defenses • Age o Must be 18 years of age o Exception: for necessities § Food, shelter, clothing, medical care § Used car ( depends) o Court only allows for reasonable value: can restructure the contract for equitable amount • Illegality o Any illegal term in the contact is unenforceable • Mental incapacity o Court decides whether the party got involved in the contract knowing that they might get taken advantage of (intoxication) • Duress o When you’re in a position where you can’t negotiate because you’re pressured by a condition or circumstance o Court may issue relief • Undue influence o Based on a relationship between the parties • Mistake o Only a bi-lateral mistake: When both parties entered the contract under a mistake o If the parties were mistaken going into the contract, the court will nullify the contract • Misrepresentation o False representation of material fact o Actual reliance must be shown o May be intentional or negligence o Can be unintended but implied o Failure to represent is included in misrepresentation • Fraud o Intentional misrepresentation o Bad faith o Consumer Protection Act or common law -> possible punitive damages § Penalties: double or triple the amount in contract and attorney’s fees • Unconscionable contract o Elements § Business to consumer transaction (can’t be between two businesses or two consumers) § Disparity in the level of sophistication between the parties (intended to protect people who are inexperienced in dealing with the market) § Contract terms are patently unfair o If these elements are met, the court will step in and nullify the contract Using these defenses can leverage your bargaining terms while negotiating. April 14, 2016 Parol Evidence Rule • Oral representation; cannot modify written contract • Exceptions o Oral clarification of meaning of terms o Incomplete contract – oral said regarding: additional terms Statute of Frauds • Contracts for sale of land • Contracts that by its terms cannot be performed within a year o You can’t put a 10-year lease for a building on oral contract • Contract for Indemnification • Contract for sale of goods over $500 under UCC (Uniform Commercial Code) Discharge of Contract • Performance o If the parties completed their performance o “Substantial performance” is sufficient • Breach o Breach of contract will discharge the non-breaching party § You’re not expected to keep performing if the other side doesn’t perform what he’s supposed to do o Anticipatory Breach § Demand reassurance of the other party’s ability to perform § Write a letter to demand reassurance despite the bankruptcy -> give information about bankruptcy/insurance in case the contract doesn’t go through § Other party must provide reasonable assurance § Failure to provide the assurance means that you can walk away from the contract (failure to provide assurance constitutes breach of contract) • Agreement of Parties o Novation: Alternate contract o Accord; Satisfaction: giving something of value to satisfy any remaining obligation regarding the contract o Novation: substitution of old contract or replacing with a new contract • Failure of a Condition o Condition precedent o Condition subsequent • Impossibility of Performance o Parties may not be responsible

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Chapter 1.2, Problem 32 is Solved
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Textbook: Linear Algebra with Applications
Edition: 4
Author: Otto Bretscher
ISBN: 9780136009269

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Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This