Calculus With Tech I

1. Calculus With Tech I Instructor: Dr. Chekad Sarami

2. What is calculus? • Calculus is a branch of mathematics that deals with rates of change. Modern Calculus began with Newton and Leibnitz in the 17th century. • Today it is used extensively in many areas of science. Basic ideas of calculus include the idea of limit , derivative , and integral .

3. Examples • The derivative of a function is its instantaneous rate of change, with respect to something else. Thus, • the derivative of height , (with respect to position) is slope ; • the derivative of position , (with respect to time) is velocity ; and • the derivative of velocity (with respect to time) is acceleration.

4. Why is calculus Extremely important? • In the sciences, many processes involving change, or related variables, are studied. • If these variables are linked in a way that involves chance, and significant random variation, statistics is one of the main tools used to study the connections. • In cases where a deterministic model is at least a good approximation, calculus is a powerful tool to study the ways in which the variables interact. • Situations involving rates of change over time, or rates of change from place to place, are particularly important examples.

5. Applications • Physics, astronomy, mathematics, and engineering make particularly heavy use of calculus; • it is difficult to see how any of those disciplines could exist in anything like its modern form without calculus. • Biology, chemistry, economics, computing science, and other sciences use calculus too. • Many faculties of science therefore require a calculus course from all their students; in other cases you may be able to choose between, say, calculus, statistics, and computer programming.

6. Resources • http://www.stewartcalculus.com/media/3_home.php • http://archives.math.utk.edu/visual.calculus/index.html • http://web.rollins.edu/~dchild/calcAssist/ • http://www.brookscole.com/cgi-wadsworth/course_products_wp.pl?fid=M20bI&product_isbn_issn=0534409865&discipline_number=1

7. Section 1.1 Functions and Models

8. Let X and Y be two nonempty sets of real numbers. A function from X into Y is a relation that associates with each element of X a unique element of Y. The set X is called the domain of the function. For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

9. f x y x y x X Y RANGE DOMAIN

10. Example: Which of the following relations are function? {(1, 1), (2, 4), (3, 9), (-3, 9)} A Function {(1, 1), (1, -1), (2, 4), (4, 9)} Not A Function

11. Functions are often denoted by letters such as f, F, g, G, and others. The symbol f(x), read “f of x” or “f at x”, is the number that results when x is given and the function f is applied. Elements of the domain, x, can be though of as input and the result obtained when the function is applied can be though of as output. Restrictions on this input/output machine: 1. It only accepts numbers from the domain of the function. 2. For each input, there is exactly one output (which may be repeated for different inputs).

12. For a function y = f(x), the variable x is called the independent variable, because it can be assigned any of the permissible numbers from the domain. The variable y is called the dependent variable, because its value depends on x. The independent variable is also called the argument of the function.

13. Example: Given the function Find: f (x) is the number that results when the number x is applied to the rule for f. Find:

14. The domain of a function f is the set of real numbers such that the rule of the function makes sense. Domain can also be thought of as the set of all possible input for the function machine. Example: Find the domain of the following function: Domain: All real numbers

15. Example: Find the domain of the following function: Example: Find the domain of the following function:

16. The domain of the function is Example: Express the area of a circle as a function of its radius. The dependent variable is A and the independent variable is r.

17. Four ways to represent a Function • verbal • numerical • visual • algebraic http://archives.math.utk.edu/visual.calculus/0/functions.11/index.html

18. The graph of f(x) is given below. y (2, 3) 4 (10, 0) (4, 0) 0 x (1, 0) (0, -3) -4

19. What is the domain and range of f ? Domain: [0,10] Range: [-3,3] Find f(0), f(4), and f(12) f(0) = -3 f(4) = 0 f(12) does not exist since 12 isn’t in the domain of f

20. A function f is even if for every number x in its domain the number -x is also in the domain and f(x) = f(-x). A function is even if and only if its graph is symmetric with respect to the y-axis. A function f is odd if for every number x in its domain the number -x is also in the domain and -f(x) = f(-x). A function is odd if and only if its graph is symmetric with respect to the origin.

21. y x (0,0) y x Example of an Even Function. It is symmetric about the y-axis Example of an Odd Function. It is symmetric about the origin (0,0)

22. Determine whether each of the following functions is even, odd, or neither. Then determine whether the graph is symmetric with respect to the y-axis or with respect to the origin. a.) Even function, graph symmetric with respect to the y-axis.

23. b.) Not an even function. Odd function, and the graph is symmetric with respect to the origin.

24. A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2). A function f is constant on an open interval I if, for any choice of x in I, the values of f(x) are equal.

25. y (2, 3) 4 (4, 0) 0 (1, 0) x (0, -3) (10, -3) (7, -3) -4 Determine where the following graph is increasing, decreasing and constant. Increasing on (0,2) Decreasing on (2,7) Constant on (7,10)

26. Section 1.2 Mathematical Models: A Catalog of Essential Functions

27. y (0,0) x The following library of functions will be used throughout the text. Be able to recognize the shape of each graph and associate that shape with the given function. y (0,c) The Constant Function x The Identity Function

28. y x (0,0) y x The Square Function The Cube Function (0,0)

29. y x (0,0) y (1,1) x (-1,-1) The Square Root Function The Reciprocal Function

30. y x y x The Absolute Value Function (0,0) The Cube Root Function

31. When functions are defined by more than one equation, they are called piecewise- defined functions. Example: The function f is defined as: a.) Find f (1) = 3 Find f (-1) = (-1) + 3 = 2 Find f (4) = - (4) + 3 = -1

32. y 3 2 1 x 1 2 3 b.) Determine the domain of f Domain: in interval notation or in set builder notation c.) Graph f

33. y 3 2 1 x 1 2 3 d.) Find the range of f from the graph found in part c. Range: in interval notation or in set builder notation

34. Power Functions

35. A polynomial function is a function of the form where an , an-1 ,…, a1 , a0are real numbers and n is a nonnegative integer. The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears.

36. Example: Determine which of the following are polynomials. For those that are, state the degree.