# MATA31 Fall 2013 Knowledge Summary

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University of Toronto Scarborough

Mathematics

MATA31H3

Zohreh Shahbazi

Fall

Description

KNOWLEDGE SUMMARY: MATA31 – Calculus I for Mathematical
Sciences (Fall 2013)
All sections and pages cited refer to the course textbook, Single Variable Calculus: Early
Transcendentals, 7E by James Stewart.
< KNOWLEDGE SUMMARY >
FUNCTIONS (CH. 1)
Definition of a function, domain & range: see p. 10
Vertical Line Test: A curve in the -plane is the graph of a function of iff any vertical line
crosses the curve only once.
Absolute value of a number :
A function is increasing [decreasing] on an interval iff :
See pp. 27-32 for a “catalog” of important functions.
Recall that . An important property of sine and cosine functions is that
they are periodic (with period ). Periodic functions are s.t. , where is the
period.
Combinations of functions:
Exponential functions are discussed in §1.5. A simple way to remember exponent laws (p. 53) is
by using the mnemonic “MADSEEM” – Multiplying exponentials? Add the exponents. Dividing
exponentials? Subtract the exponents. Exponent to Exponent: Multiply them.
A function is 1-1 if . In fact, is 1-1 iff any horizontal line crosses its
graph only once. (Horizontal Line Test)
Inverse functions are discussed on pp. 60-62.
Logarithmic (i.e. inverse exponential) functions in particular are discussed on pp. 62-65.
Note that and .
Inverse trigonometric functions are discussed on pp. 67-69.
LIMITS & DERIVATIVES (CH. 2)
The provisional definition of a limit is given on p. 87. In precise terms (p. 110):
iff , s.t. . The similar definition of a 1-sided
limit is given on p. 92 and p. 113.
1 KNOWLEDGE SUMMARY: MATA31 – Calculus I for Mathematical
Sciences (Fall 2013)
Relation between 1-sided and 2-sided limits:
The definition of an infinite limit is given on pp. 93-94 and pp. 115-116. One-sided infinite limits
indicate vertical asymptotes of a function (see p. 94). Note that .
Limit Laws:
Define functions .
Basic Algebraic Limits
1. (Constant Law)
2. (Constant Multiple Law)
3. (Sum/Difference Law)
4. (Product Law)
5. (Quotient Law)
6. (Absolute Value Law)
Consequent Limits
1.
2.
(Power Law)
3.
(Root Law)
Direct Substitution Property: For all points in the domain of a polynomial or rational function
, .
Remarks:
1. (In fact, this statement with “=”
replaced by “≤” also holds.)
2.
Squeeze Thm.:
2 KNOWLEDGE SUMMARY: MATA31 – Calculus I for Mathematical
Sciences (Fall 2013)
Limits at infinity, which indicate horizontal asymptotes are discussed on pp. 130-140.
N.B.: .
Remarks:
1. Let . If , then DNE.
2. As , the limit of a polynomial in is the same as that of its term containing the
highest power of .
A function is continuous at :
iff
iff
iff
We say is continuous on (its domain) if it is continuous at all points .
If f and g are functions continuous at a point , then so are the following:
1.
2.
3.
Continuity of “Standard” Functions (Direct Substitution Property restated!):
1. Polynomial & rational functions are continuous on their domains.
2. Root & (inverse) exponential functions are also continuous on their domains.
In particular, a function is discontinuous (i.e. not continuous) at if:
or (infinite discontinuity)
(removable discontinuity)
(jump discontinuity)
Thm.:
Thm. (Continuity of Compositions): If is cts at and is cts at , then is cts at .
Intermediate Value Thm.: If is cts on and (or ),
then s.t. .
The derivative is defined on p. 146.
Remark: (so )
Thm.: If a function is differentiable at , then it is also cts at .
3 KNOWLEDGE SUMMARY: MATA31 – Calculus I for Mathematical
Sciences (Fall 2013)
Lemma:
DIFFERENTIATION RULES (CH. 3)
Assume & are differentiable functions and that .
1. Constant Rule:
2. Sum/Difference Rule:
3. Product Rule:
4. Power Rule: (Special case when n 1 )
5. Quotient Rule: (Special case when f 1 )
6. Chain Rule:
Remarks:
1. [Special case: ]
2. [Sp

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