MATA32 Final Exam Analysis

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University of Toronto Scarborough
Raymond Grinnell

Final Exam Analysis: MATA32 – Calculus for Management I 1. TEST BREAKDOWN The MATA32 final exam is generally cumulative, covering 5 main topics: the 3 main topics covered on the midterm test, an expansion of the differentiation topic, and the new topic of integration. These are listed below.  Financial Mathematics  Limits  Differentiation  Derivatives & Applications (includes implicit & logarithmic differentiation, Newton’s Method, higher-order derivatives, and curve sketching)  Integrals & Applications The test is usually 180 minutes (3 hours) in length, and the format is very similar to that of the midterm test:  A) Multiple Choice Questions (usually 10-12 total)  B) Full Solution Problems (usually 7-10 total) 2. TEST STATISTICS Winter 2013 Fall2012 Old Old New Topics New Topics Content 45% Content 47% 55% 53% Winter 2012 Fall2011 New Content Old New Old 43% Topics Content Topics 57% 50% 50% Winter 2011 Fall2010 Old Old Topics New Topics New 42% Content 44% Content 56% 58% 1 Final Exam Analysis: MATA32 – Calculus for Management I 3. TOPIC SUMMARIES All sectionthand pages cited refer to the course textbook, Introductory Mathematical Analysis, 13 ed. by Haeussler, Paul, & Wood. < KNOWLEDGE SUMMARY > FINANCIAL MATHEMATICS See Midterm Exam LIMITS Analysis DIFFERENTIATION DERIVATIVES & APPLICATIONS Implicit & Logarithmic Differentiation (§12.4-12.5) An equation of the form (e.g. ) does not express explicitly as a function of . That is to say, in such equationis expressed implicitly as a function of . To find , we use the technique of implicit differentiation (described on p. 557). Logarithmic differentiation (described on p. 561) is a technique used to ease the process of differentiating functions involving products, quotients, or powers – especially those of the form , where and are both functions of , and thus variable as opposed to constant. Newton’s Method (§12.6) Finding the roots of a non-polynomial function or even a polynomial function of degree >2, can often prove to be a tedious (or impossible) task by standard methods. Hence, we may settle for approximate roots. Newton’s method (discussed on pp. 564-565) is often useful in this regard. Given an initial approximation (or “seed”) , successive approximations of roots of the equation are given by the following iteration formula, provided that is differentiable: where Higher-Order Derivatives (§12.7) See p. 568. Convenient notation for derivatives: 2 Final Exam Analysis: MATA32 – Calculus for Management I (“ th derivative of ”, particularly handy for derivatives of order higher than 3) Curve Sketching (Ch. 13) A nice chapter summary can be found on pp. 620-621. A function is increasing [decreasing] on an interval iff : Extrema are maxima and minima. Relative (or local) extrema and the 1 -derivative test are discussed in §13.1. Extreme Value Thm. (EVT): If a function is continuous on , then s.t. . (I.e., has both an absolute/global maximum and minimum value over .) The closed-interval method
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