Final Exam Analysis: MATA32 – Calculus for Management I

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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

Old Topics

45%

New Content

55%

Winter 2013

Old Topics

47%

New Content

53%

Fall 2012

Old Topics

57%

New Content

43%

Winter 2012

Old Topics 50%

New Content

50%

Fall 2011

Old Topics 44%

New Content

56%

Winter 2011

Old Topics

42% New

Content 58%

Fall 2010

Final Exam Analysis: MATA32 – Calculus for Management I

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3. TOPIC SUMMARIES

All sections and pages cited refer to the course textbook, Introductory Mathematical Analysis, 13th ed. by Haeussler, Paul, & Wood. < KNOWLEDGE SUMMARY > FINANCIAL MATHEMATICS LIMITS 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 equations, is 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:

See Midterm Exam

Analysis

Final Exam Analysis: MATA32 – Calculus for Management I

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(“ 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 1st-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 can be used to find absolute extrema of a function continuous on :

1. Find the critical numbers of (i.e., all s.t. or DNE). 2. Find , , and . Then .

Concavity and inflection points are discussed in §13.3, and the 2nd-derivative test is introduced in §13.4. Asymptotes (vertical, horizontal, and slant) are discussed in §13.5. Applied maxima & minima problems appear in §13.6. Profit maximization is one such type of problem, arising frequently in economics. INTEGRALS & APPLICATIONS Integration (§14.2-14.7, §14.9-14.10, §15.1) The antiderivative of a function is a function s.t. – i.e., . We have that .

Final Exam Analysis: MATA32 – Calculus for Management I

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Some elementary integration formulas are listed on p. 633 (Table 14.1). A few other rules are mentioned in §14.4. The definite integral is formally discussed in §14.6. Fundamental Thm. of Calculus (FTC): Suppose the function is integrable.

1. Define by

. Then whenever f is

continuous on .

2. Net Change Thm.: If s.t. , then

.

If , then

.

Some other important properties of the definite integral are presented on p. 662. The area between the curves and and between and is:

So we subtract the “lower curve” from the “upper curve”, according as or .

Application: Consumer’s Surplus (CS) & Producer’s Surplus (PS) Suppose the market for a product is at equilibrium, so that the supply curve and the demand curve intersect. If is the point of intersection, then:

o

, and

o

.

Formula for Integration by Parts:

< SUMMARY OF POTENTIAL QUESTIONS > “Derivatives & Applications” as presented in this document is merely a continuation of the differentiation topic, as covered on the midterm exam. Two new techniques are introduced: implicit and logarithmic differentiation. Newton’s method, an important part of numerical analysis, is also introduced. Curve sketching largely deals with concepts and techniques learned in high school mathematics; it is given a more formal treatment in MATA32. Integration is an entirely new topic to most first-year university students, and is handled “delicately” in MATA32. It is important to recognize that while

Final Exam Analysis: MATA32 – Calculus for Management I

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integration and differentiation are just inverse processes, finding integrals can often be quite tedious in comparison to finding derivatives. It is necessary to get as much practice with integrals as possible before the final exam. The test statistics presented earlier suggest that more than 40% of the final exam will likely be allocated to content previously encountered on the midterm test. Thus, it is wise to review the Midterm Exam Analysis document and associated prep videos if the midterm exam proved to be challenging and/or the relevant topics are not fresh in your mind. Observe that some of this “old content” fits in quite well with the new content that will appear on the final exam. Thus, it is quite possible for a single final exam question to incorporate both old and new concepts that are somehow related. < RELATED QUESTIONS FROM PAST EXAMS > Past final exams from the following years can be found on your course website:

Winter 2013

Fall 2012

Winter 2012

Fall 2011

Winter 2011

Fall 2010 On the Midterm Exam Analysis document, questions from the more recent tests were color coded by topic and listed in a chart, for your convenience. At this point, it should be fairly easy to identify exam questions by topic, so one can either sort the past final exam questions accordingly and work on them selectively or just attempt each past exam from start to finish!