Lecture 1 admin & representing fcts
- 1. 1 Math 138 Section 003 Professor Brown Fall 2014
- 2. 2 Prof Brown Contact Info ronbrown@njit.edu (best) 973-642-4096 (I don’t check that often) Office Hours: Tuesday 10 to 1 Wednesday 11:30 to 12:30 CULM 212A (back of the adjunct office)
- 3. 3 Course Information Start with Moodle page if you have any questions Class Expectations Academic Integrity Policy Attendance Policy Quiz Policy Exam Policy
- 4. 4 Grading Quizzes 15% Midterm 1 25% Midterm 2 25% Final 35%
- 5. 5 Resources Textbook My office hours Math Learning Center (CULM 214) Internet (Kahn Academy, etc)
- 6. Four Ways to Represent 1.1 a Function
- 7. 7 What is a Function? Functions arise whenever one quantity depends on another. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f (x) is the value of f at x and is read “f of x.” The range of f is the set of all possible values of f (x) as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable.
- 8. 8 What is a Function? It’s helpful to think of a function as a machine Domain Independent Variable Range Dependent Variable or y Key Idea – each input has only one output
- 9. 9 Representations of Functions There are four possible ways to represent a function: verbally (by a description in words) numerically (by a table of values) algebraically (by an explicit formula) visually (by a graph)
- 10. 10 Verbal Representation • Revenue is $10 for every unit sold • Force required to stretch/compress a spring is proportional to the distance spring is stretched/compressed • Voltage across a capacitor decays exponentially with a time constant RC
- 11. 11 Numerical Representation The human population of the world P depends on the time t. The table gives estimates of the world population P(t) at time t, for certain years. For instance, P(1950) » 2,560,000,000 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t.
- 12. 12 Algebraic Representation The area of a circle A = p r 2 Height of projectile: h(t) = -16t2+ vt + h0
- 13. 13 Graphical Representation The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5. Figure 5
- 14. Example 1 – Reading Information from a Graph The graph of a function f is shown in Figure 6. 14 Figure 6 The notation for intervals is given in Appendix A. (a) Find the values of f (1) and f (5). (b) What are the domain and range of f ?
- 15. 15 Example 1 – Solution
- 16. Vertical Line Test The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test. 16
- 17. 17 Representations of Functions The reason for the truth of the Vertical Line Test can be seen in Figure 13. Figure 13 If each vertical line x = a intersects a curve only once, at (a, b), then exactly one functional value is defined by f (a) = b. But if a line x = a intersects the curve twice, at (a, b) and (a, c), then the curve can’t represent a function because a function can’t assign two different values to a.
- 18. 18 Vertical Line Test Draw a graph that fails the vertical line test at x=3 Draw a graph that passes the vertical line test.
- 19. 19 Finding Domains • Sometimes explicitly given • Sometimes implied – word problem describing area of a circle as a function of radius, implies radius is positive number • Exclude values that would “break” the function: • No division by zero • No even roots of negative numbers • No logs of non-positive numbers
- 20. 20 Finding Domains - Examples
- 21. 21 Finding Domains - Examples
- 22. 22 Finding Domains - Examples
- 23. 23 Finding Domains - Examples