Calculus

1. Calculus Guanying Sun(孙冠颖) Calculus 厦门鼓浪屿

2. Textbooks and References • 1. Thomas George B., Finney Ross L., Weir, Maurice D. and Giordano, Frank R. Thomas’ Calculus (Tenth Edition), Pearson Education Asia Limited and Higher Education Press, 2004. (Textbook) • 2. Stewart, J., Calculus(Fifth Edition), Brooks Cole, 2002. • 3. Larson R., Hostetler Robert P. and Edwards Bruce H., Calculus(Eighth Edition), Brooks Cole, 2005. • 4. 同济大学数学系,《高等数学（第六版）》，高等教育出版社，2007. • Lecture slides will be uploaded to http://e.ncut.edu.cn/ after each lecture Calculus

3. Grading • Lecture Attendance 20％ of grade • Homework and Quiz 20％ of grade • Final Exam 60％ of grade closed-book Calculus

4. Contact Information • ★Office Hour:Thursday 15:45PM-17:45PM • ★ Office: West 204, 4th Academic Building • ★ Email:gysun@ncut.edu.cn • ★ Tel:15110204523 Calculus

5. Any questions on this course? Calculus

6. What is Calculus? Calculus

7. Origin of calculus • The word Calculus comes from the Greek name for pebbles. • Pebbles were used for counting and doing simple algebra… Calculus

8. American Heritage Dictionary Pathology(病理学). An abnormal concretion in the body, usually formed of mineral salts and found in the gallbladder, kidney, or urinary bladder, for example. ——结石 Calculus

9. American Heritage Dictionary Dentistry. A hard yellowish deposit on the teeth, consisting of secretions and food particles deposited in various salts, such as calcium carbonate. Also called tartar.——牙垢 Calculus

10. American Heritage Dictionary • Mathematics. ——微积分 • The branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables. • A method of analysis or calculation using a special symbolic notation. • The combined mathematics of differential calculus and integral calculus. Calculus

11. www.wikipedia.com ♦Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. ♦It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus

12. Calculus has widespread uses in science, economics, and engineering and can solve many problems. Calculus

13. The History Of Calculus

14. In the earliest years, integral calculus was being used as an idea, but was not yet formalized into a system. Calculating volumes and areas can be traced to the Egyptian Moscow papyrus (1820 BC). Ancient History

15. Greek mathematician Eudoxus (408-355 BC) used the method of exhaustion, a precursor to limits, to calculate area and volume Archimedes (287-212 BC) continued Eudoxus’ idea and invented heuristics, similar to integration, to calculate area. Ancient Greeks

16. In about 1000 AD, Islamic mathematician, Ibnal-Haytham(Alhacen) derived a formula for the sum of the fourth powers of an arithmetic progression, later used to perform integration. In the 12th century, Indian mathematician Bhaskara II developed an early derivative. He described an early form of what will later be “Rolle’s Theorem” Also in the 12th century, Persian mathematician Saraf al-Din al-Tusi discovered the derivative of a cubic polynomial Medieval History

17. Bonaventure Cavalieri （卡瓦列里 ）argued that volumes be computed by the sums of the volumes of cross sections. (This was similar to Archimedes’s). However, Cavalieri’s work was not well respected, so his infinitesimal quantities were not accepted at first. Modern History

18. Formal study combined Cavalieri’s infinitesimal quantities with finite differences in Europe. This was done by John Wallis(沃利斯), Isaac Barrow（伊萨克·巴罗）, and James Gregory （格雷戈里） Barrow and Gregory would later prove the 2nd Fundamental Theorem of Calculus in 1675. Modern History

19. Isaac Newton (English) is credited with many of the beginnings of calculus. He introduced product rule, chain rule and higher derivatives to solve physics problems. He replaced the calculus of infinitesimals with geometric representations. He used calculus to explain many physics problems in his book Principia Mathematica（自然哲学的数学原理）, however he had developed many other calculus explanations that he did not formally publish. Newton…

20. Gottfried Wilhelm Leibniz (German) systemized the ideas of calculus of infinitesimals. Unlike Newton, Leibniz provided a clear set of rules to manipulate infinitesimals. Leibniz spent time determining appropriate symbols and paid more attention to formality. His work leads to formulas for product and chain rule as well as rules for derivatives and integrals. …and Leibniz

21. There was much controversy over who (and thus which country) should be credited with calculus since both worked at the same time. Newton derived his results first, but Leibniz published first. Newton vs. Leibniz

22. Newton claimed Leibniz stole ideas from unpublished notes written to the Royal Society. This divided English-speaking math and continental math for many years. Newton vs. Leibniz

23. Today it is known that Newton began his work with derivatives and Leibniz began with integrals. Both arrived at the same conclusions independently. The name of the study was given by Leibniz, Newton called it “the science of fluxions”. Newton vs. Leibniz

24. There have been many contributions to build upon Newton and Leibniz. Calculus was put on a more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass Calculus has also been generalized for the Euclidean and complex space. Since then…

25. Short Review of Pre-Calculus Calculus

26. 1.Functions and Graphs Domain and Range Functions have an independent variable (often x) and a dependent variable (often y). domain: The set of all possible x values for a function. range: The set of all possible y values for a function.

27. Example: Find the domain and range of: In this case, x cannot equal zero. The domain is: or This is set builder notation. We usually use interval notation. These are open intervals (with parentheses) because we don’t count the zero. Also notice the symbol for union.

28. or Looking at the graph, we see that we can have every y value except zero, so the range is:

29. Another example: Find the domain and range of: Since we cannot take the square root of a negative number, the domain is: This is a half-open interval. We use a square bracket on the left boundary point because we count the zero. (The interval is closed at zero.) When infinity is a boundary, it is always an open interval on that end, with parentheses.

30. Another example: Find the domain and range of: Since we cannot take the square root of a negative number, the domain is: Looking at the graph, we see that there are no negative y values, so the range is:

31. Symmetry When we graph the function , we see that it has y-axis symmetry. When we square the x, any negative sign cancels out, so changing the sign of x does not change the y value. Any polynomial function with only even exponents behaves the same way, and has y-axis symmetry. This is , which has an even exponent.

32. Any function with y-axis symmetry is called an even function. For example, is an even function. So is .

33. A polynomial function with only odd exponents has origin symmetry. Changing the sign of x changes the sign of y. In other words, if (x,y) is on the graph, so is (-x,-y).

34. Any function with origin symmetry is called an odd function. For example, is an odd function. Polynomial functions with exponents that are both even and odd have no symmetry.

35. Of course, a graph with x-axis symmetry is not a function at all! Fails the vertical line test!

36. Piecewise Functions While many functions can be defined by a single formula, others are defined by applying different formulas to different parts of their domains. Example: Just graph each piece separately, for each part of the domain.

37. Example: Write a piecewise function for the graph at right: There are two pieces to this graph, so we need two equations. The equation for the left hand piece is: For the right hand piece, the equation is: This simplifies to: Paying attention to the open and closed circles, we get the piecewise function:

38. How to shift a graph The rules for shifting, stretching, shrinking, and reflecting the graph of a function make it easier to sketch functions by hand. Since we will be frequently using graphs in our study of calculus, we will do a quick review of those rules. If we know how to graph the “parent” graph of a function, then we can modify that graph to get the one we want.

39. Example: Adding a positive number at the end moves the graph up.

40. Example: Adding a constant to x inside the parentheses moves the graph to the left. The horizontal changes happen in the opposite direction to what you might expect.

41. Example: Placing a coefficient in front of the function causes a vertical stretch. In this case, the graph goes up twice as fast.

42. Example: If the coefficient is negative, then the graph is reflected about the y-axis.

43. Example: Placing a coefficient inside the function in front of the x causes a horizontal shrink. In this case, the graph expands horizontally half as fast. The horizontal changes happen in the opposite direction to what you might expect.

44. Example: Clearing the parentheses: In this case, a horizontal shrink is the same as a vertical stretch, but this is not always true.

45. Example: If the coefficient inside the function in front of the x is negative, you get a reflection about the y axis. In this case, since we started with an even function, we cannot see the reflection. Let’s look at an odd function.

46. Example: Placing a negative coefficient inside the function in front of the x causes a reflection about the y-axis.

47. is a stretch. is a shrink. To summarize the rules for transformations of graphs: Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. The horizontal changes happen in the opposite direction to what you might expect.

48. 2. Exponential functions Calculus

49. Calculus

50. 4.Inverse functions and Logarithms Calculus