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Final

# MATA30 Final Exam Analysis

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Department
Mathematics
Course
MATA30H3
Professor
Sophie Chrysostomou
Semester
Fall

Description
Final Exam Analysis: MATA30 – Calculus I for Biological & Physical Sciences 1. TEST BREAKDOWN The MATA30 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.  Functions  Limits & Derivatives  Differentiation Rules  Applications of Differentiation  Integration The test is usually 170 minutes (almost 3 hours) in length, consisting of 2 parts as follows:  A) Multiple Choice Questions ousually about 10 questions which test your understanding of key concepts and/or involve straight-forward calculations (e.g. direct calculations of limits or derivatives)  B) Full Solution Problems ooften 5-7 questions, each of which have multiple parts (a, b, c, d, etc.) oalmost all the questions are calculation-based (both direct calculations and applications), with the possible exception of 1 or 2 questions requiring a proof of a trig identity (just as in the midterm test). oSome questions may ask you to state a certain definition or theorem In studying for the final exam, it is highly recommended that you thoroughly review your lecture notes, although the presentation of concepts in the textbook is very good. 2. TEST STATISTICS Fall 2008 Final Exam Functions 14% Integration Limits 44% 14% Differentiation 14% Applicationsof Differentiation 14% 1 Final Exam Analysis: MATA30 – Calculus I for Biological & Physical Sciences 3. TOPIC SUMMARIES All sections and pages cited refer to the course textbook, Single Variable Calculus: Early Transcendentals, 7E by James Stewart. < KNOWLEDGE SUMMARY > FUNCTIONS (CH. 1) See Midterm Exam LIMITS & DERIVATIVES (CH. 2) DIFFERENTIATION RULES (CH. 3) Analysis APPLICATIONS OF DIFFERENTIATION (CH. 4) Extrema are maxima and minima, which come in two forms: global (or absolute) and local (or relative). These are defined on p. 274. 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 .) A critical number of a function is a number s.t. or DNE. The value is called the critical value of . 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 . Rolle’s Thm.: If a function is continuous on , differentiable on , and , then s.t. . Corollary: If is continuous on , differentiable on , and has roots in , then has roots in . Mean Value Thm. (MVT): If a function is continuous on and differentiable on , then s.t. . Corollary 1 (Constant Function Thm.): (Replacing by yields a more general result.) 2 Final Exam Analysis: MATA30 – Calculus I for Biological & Physical Sciences Corollary 2 (Simplified Increasing/Decreasing Test): If on an interval, then is increasing [decreasing] on that interval. Corollary (1 Derivative Test): Suppose is differentiable on and that is a critical number of . 1. If , then has a local max. [min.] at . 2. If , then has no extremum at Concavity, inflection points, and the 2 Derivative Test are discussed on pp. 293-295. L’Hospital’s Rule is stated on p. 3
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