2. FUNDAMENTAL PHYSICS (I)

3. The Textbook FUNDAMENTALS OF PHYSICS (Sixth Edition) David Halliday, Robert Resnick, and Jearl Walker

4. Reference Books UNIVERSITY PHYSICS by George B. Arfken, David F. Griffing, Donald C. Kelly, and Joseph Priest PHYSICS (First Edition) By Hans C. Ohanian 《新概念物理教程  力学》 赵凯华 罗蔚茵 《力学》 漆安慎 杜蝉英

5. About this course Bi-language is used to improve the skill of students ’ reading physical texts. The content and the schedule of the teaching may be varied according to the situation we will have. In order to give students some introductory practice in scientific research, a scientific paper, with arbitrary physical content, is required before the end of this term. Every student has to attend two scientific seminars given by teachers in our college and some invited physicists. Assignment is to be handed in and marked once a week. For those who write less than two thirds of the total homework their qualification of attending the final examination will be deprived.

6. My personal information Name:Xiaobing Chen Office:209 Physics Building Lab:619 Physics Building Tel: 797-5511(O), -5489(L), 7891906(H) Email: xbchen@yzu.edu.cn or xbchen@msn.com xbchen@yzu.edu.cnxbchen@msn.com

7. Any comments, questions and suggestions, from everyone, by any manner, are cordially welcomed! Any comments, questions and suggestions, from everyone, by any manner, are cordially welcomed! Electronic communication are welcomely encouraged! Electronic communication are welcomely encouraged!

8. APPENDIX : CALCULUS A.1 Functions, Derivatives, and Differentials A.1.1 Variables and functions 1. The definition of a function where x is a variable and y is called the function of variable x. Different alphabets, such as “g”, “F”, “  ”, and so on, can be used to distinguish various functions of variable x.

9. If the function, y has a definite value of y o = f(x o ) as the variable x taken as x o, it is said that the function y is defined at x o. The set of all defined values of x is called the domain of the function. The set comprising all the values assumed by as x takes on all possible value in its domain is called the range of the function f.

10. 2. Basic functions (1) Constant functions: (C is a constant) (2) Power functions:(  is a real number) (3) Exponential functions:(a > 0, a  1) (4) Logarithmic functions: (a > 0, a  1) (5)Trigonometric functions:

11. (6) Inverse trigonometric functions: 3. Compound functions:

12. A.1.2 Derivatives 1. Limit of a function The function f has the limit L as x approaches a, written if the value f(x) can be made as close to the number L as we please by taking x sufficiently close to (but not equal to) a.

13. Example 1: Consider the values of the function as x gets very close to x = 2. xf(x)f(x) 18 1.59.5 1.910.7 1.9910.97 1.99910.997 2.00111.003 2.0111.03 2.111.3 2.512.5 314 Note that in this example we might have guessed that the limit of as x approaches 2 is 11 by observing that f(2) = 11.

14. Example 2: Consider the values of the function As x approaches x =  1. xf(x)f(x) 22 0  1.5 0.5  1.1 0.9  1.01 0.99  1.001 0.999  0.999 1.001  0.99 1.01  0.9 1.1  0.5 1.5 02 Note that  1 is not the domain of f(x) because the denominator is zero when x =  1. However, we are not interested in x =  1, but only in x-values close to  1. We may factor the numerator and get

15. Limit theorems are as follows: A. for all constants c. B. for all positive numbers n. C. for all constants c. D. E.

16. F. provided G. for all positive numbers r. Rational function rule: where f(x) and g(x) are polynomials and g(a)  0.

17. 2. Definition of derivatives The derivative of a function f(x) at a value a is defined to be provided the limit exists. If the limit does not exist, then we say that f(x) has no derivative at a. If a function f(x) has a derivative at x = a, we say that f(x) is differentiable at a.

18. If f(x) is differentiable for all a in its domain, then f(x) is said to be a differentiable function. The process of finding the derivative of a function is differentiation. Definition of the derivative Other notations for the derivative of f(x) are

19. 3. Derivative rules: A. B. C. D. If is the inverse function of, then provided E. If, then

20. 4. Some basic derivative formulas: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

21. (13) (14) The significance of a derivative is the slope of f(x) at a value x

22. A.1.3 Increasing and decreasing functions A function f(x) is increasing on an interval if f(x 2 ) > f(x 1 ) for every pair of values x 1 and x 2 in the interval with x 2 > x 1. A function is decreasing on an interval if f(x 2 ) x 1 If f(x) is an increasing function on an interval and f’(x) exists there, then f’(x)  0 on that interval. If f(x) is an decreasing function on an interval and f’(x) exists there, then f’(x)  0 on that interval. Any point on the graph of f(x) where f’(x) equals zero or does not exist is called a critical point. At such a point a function may switch from increasing to decreasing or from decreasing to increasing.

23. A.1.4 The second derivative: concavity Let f(x) be a function such that f’(x) exists. If the function f’(x) is also differentiable, then the “derivative of f’(x) is called the second derivative of f(x) and is denoted by f’’(x) or, sometimes, by f (2) (x). Continuing in this way, f (n) (x) denotes the nth derivative of f(x) and is defined to be the derivative of f (n  1) (x). In general, derivatives other than f’(x) are called higher- order derivatives. The second derivative of f(x) may be denoted as.

24. When the curve is “bending downward” in this manner, we say that the graph of the function is concave downward. The graph of a function f(x) is concave downward whenever f’’(x) < 0. Similarly, if the graph of a function f(x) is “bending upward,” the graph is said to be concave upward; this occurs whenever f’’(x) > 0. A point on the graph of a function at which the function changes from concave upward to concave downward, or vice versa, is called an inflection point. An inflection point can occur only where f’’(x) = 0 or where f’’(x) is not defined.

25. A.1.5 Derivative tests for relative maxima and relative minima A function f(x) has a relative maximum f(a) at the point (a, f(a)) if f(a)  f(x) for all x in some open interval containing a. In other words, f(a) is a maximum for the function if we consider only values of x near a. A function f(x) has a relative minimum f(a) at the point (a, f(a)) if f(a)  f(x) for all x in some open interval containing a. The first derivative test: The second derivative test:

26. A.1.6 Absolute maxima and absolute minima The absolute maximum of a function is a value f(a) such that f(a)  f(x) for all values of x in the domain of f(x). Similarly, the absolute minimum of a function is a value f(a) such that f(a)  f(x) for all values of x in the domain of f(x). Find absolute maxima and absolute minima: Let f(x) be a continuous function on the finite closed interval [b, c]. Find all critical values of a in (b, c). Of the values f(b), f(c), and these f(a)’s, the largest is the absolute maximum of f(x) on [b, c] and the smallest is the absolute minimum of f(x) on [b, c].

27. A.1.7 Absolute maxima and absolute minima We may use the differential of a function to calculate approximately the increasing of the function when the independent variable is increasing. A differential of a function is defined as The significance of differential of a function is that an increasing of a function is approximately equal to the differential of the function:

28. A.1 Integration A.2.1 Antidifferentiation An antiderivative of a function f(x) is a function F(x) such that F’(x) = f(x). If F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative of f(x) for any constant C. The requirement that the graph of an antiderivative must passes through a given point (a, b) is called an initial condition. If f(x) has an antiderivative, then we let the symbol Indefinite integral

29. A.2.2 Integration rules A. B. C. D. (Integration by parts formula).

30. A.2.3 The definite integral To find the area under the graph of f(x) between two vertical lines x = a and x = b, we divide the interval [a, b] into n subintervals. Riemann sum method

31. The definition of the definite integral of f(x): The integral is called the definite integral from a to b of f(x).

32. A.2.4 Fundamental theorem of calculus where F(x) is any antiderivative of f(x).

33. Exercises: 1. Find the derivative of each function. (1) (2) (3) (4) (5) (6)

34. (7) (8) (9) (10)

35. 2. Evaluate indefinite integrals: (1) (2) (3) (4) (5) (6)

36. 3. Calculate definite integrals: (1) (2) (3) (4) (5) (6)