Final Exam Analysis: MATA30 – Calculus I for Biological & Physical
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.
Limits & Derivatives
Applications of Differentiation
The test is usually 170 minutes (almost 3 hours) in length, consisting of 2 parts as
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
1 Final Exam Analysis: MATA30 – Calculus I for Biological & Physical
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 .
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
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