CALCULUS I Enea Sacco Welcome to Calculus I

CALCULUS I Enea Sacco

Welcome to Calculus I! 2

WELCOME TO CALCULUS I Topics/Contents Before Calculus Functions. New functions from the old. Inverse Functions. Trigonometric Functions. Inverse Trigonometric Functions. Exponential and Logarithmic Functions Limits and continuity Limits, an Intuitive Approach. Computing Limits more Rigorously. Continuity of Trigonometric, Exponential and Inverse Functions The derivative Tangent Lines and Rate of Change. The Introduction to the Techniques of Differentiation The Product and the Quotient Rule. Derivatives of Trigonometric Functions. The Chain Rule The derivative in graphing and applications Increasing, Decreasing and Concave Functions. Relative Extrema. Graphing Polynomials. Absolute Maxima and Minima. Graphing function. Applied Maximum and Minimum Problems Integration The indefinite Integral. Integration by Substitution. Integration by Parts. The Definite Integral. Applications of definite integral. The Fundamental Theorem of Calculus. Integrating Trigonometric Functions. Trigonometric Substitutions. Area Between Two Curves 3

BOOK CALCULUS EARLY TRANSCENDENTALS 9 th edition by HOWARD ANTON, IRL BIVENS, STEPHEN DAVIS. 4

EVALUATION Assiduity and attendance Homework assignments (1 every 2 weeks) Midterm Final Total 10% 30% 30% 100% 5

WHAT IS A FUNCTION? If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x. 6

COMMON WAYS OF REPRESENTING FUNCTIONS Numerically by tables Geometrically by graphs Algebraically by formulas Verbally 7

DENOTING FUNCTIONS BY LETTERS OF THE ALPHABET 8

INDEPENDENT AND DEPENDENT VARIABLES Independent variable (or argument) Dependent variable 9

EXAMPLE OF A FUNCTION age 0 0 1 20 3. 2 64 15 300 . . . 10

EXAMPLE OF A FUNCTION (2) 11

GRAPHS OF FUNCTIONS A very useful way of representing functions is through graphs. 12

THE VERTICAL LINE TEST 13

THE ABSOLUTE VALUE FUNCTION The effect of taking the absolute value of a number is to strip away the minus sign if the number is negative and to leave the number unchanged if it is nonnegative. For example 14

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PROPERTIES OF ABSOLUTE VALUES 16

PIECEWISE FUNCTIONS 17

PIECEWISE FUNCTIONS 18

EQUATION FOR A CIRCLE 19

DOMAIN AND RANGE OF A FUNCTION 20

DOMAIN AND RANGE OF A FUNCTION 21

NEW FUNCTIONS FROM OLD 22

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COMPOSITION OF FUNCTIONS 24

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INVERSE FUNCTIONS 26

INVERSE FUNCTIONS 27

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INVERSE TRIGONOMETRIC FUNCTIONS 29

EXPONENTS 30

EXPONENTS WITH FRACTIONS 31

OPERATIONS WITH EXPONENTS 32

THE EXPONENTIAL FUNCTION 33

THE EXPONENTIAL FUNCTION 34

THE EXPONENTIAL FUNCTION 35

LOGARITHMIC FUNCTIONS 36

ALGEBRAIC PROPERTIES OF LOGARITHMS 37