MATA31 Fall 2013 Knowledge Summary

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University of Toronto Scarborough
Zohreh Shahbazi

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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