1. C ALCULUS I Enea Sacco

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3. W ELCOME TO C ALCULUS I 3 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. Limits more Rigorously. Continuity. 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

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

5. E VALUATION 5 Assiduity and attendance10% Homework assignments (1 every 2 weeks)30% Midterm30% Final30% Total100%

6. W HAT 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

7. C OMMON WAYS OF REPRESENTING FUNCTIONS Numerically by tables Geometrically by graphs Algebraically by formulas Verbally 7

8. D ENOTING FUNCTIONS BY LETTERS OF THE ALPHABET 8

9. I NDEPENDENT AND DEPENDENT VARIABLES 9 Independent variable (or argument) Dependent variable

10. E XAMPLE OF A FUNCTION 10 age 00 120 3.264 15300...

11. E XAMPLE OF A FUNCTION (2) 11

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

13. T HE VERTICAL LINE TEST 13

14. T HE 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 non-negative. For example 14

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16. P ROPERTIES OF ABSOLUTE VALUES 16

17. P IECEWISE FUNCTIONS 17

18. P IECEWISE FUNCTIONS 18

19. E QUATION FOR A CIRCLE 19

20. D OMAIN AND RANGE OF A FUNCTION 20

21. D OMAIN AND RANGE OF A FUNCTION 21

22. N EW FUNCTIONS FROM OLD 22

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24. C OMPOSITION OF FUNCTIONS 24

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26. I NVERSE FUNCTIONS 26

27. I NVERSE FUNCTIONS 27

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29. I NVERSE TRIGONOMETRIC FUNCTIONS 29

30. E XPONENTS 30

31. E XPONENTS WITH FRACTIONS 31

32. O PERATIONS WITH EXPONENTS 32

33. T HE EXPONENTIAL FUNCTION 33

34. T HE EXPONENTIAL FUNCTION 34

35. T HE EXPONENTIAL FUNCTION 35

36. L OGARITHMIC FUNCTIONS 36

37. A LGEBRAIC PROPERTIES OF LOGARITHMS 37