MATH 137 MIDTERM

1. MATH 137 MIDTERM

2. 2010 Outreach Trip Summary Date Aug 20 – Sept 4 Location Cusco, Peru # Students 22 Project Cost $16,000 Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects ELT Classroom enables better job prospects More info @ studentsofferingsupport.ca/blog

3. Introduction • Arjun Sondhi • 2A Statistics/C&O • First co-op in Gatineau, QC Root beer float at Zak’s Diner in Ottawa!

4. Agenda • Functions and Absolute Value • One-to-One Functions and Inverses • Limits • Continuity • Differential Calculus • Proofs (time permitting)

5. Functions and Absolute Value REVIEW OF FUNCTIONS

6. Functions and Absolute Value • A function f, assigns exactly one value to every element x • For our purposes, we can use y and f(x) interchangeably • In Calculus 1, we deal with functions taking elements of the real numbers as inputs and outputting real numbers

7. Functions and Absolute Value • Domain: The set of elements x that can be inputs for a function f • Range: The set of elements y that are outputs of a function f • Increasing Function: A function is increasing over an interval A if • for all , the property holds. • Decreasing Function:A function is decreasing over an interval A if • for all , the property holds.

8. Functions and Absolute Value • Even Function: A function with the property that for all values of x: • Odd Function: A function with the property that for all values of x: • A function is neither even nor odd if it does not satisfy either of these properties. • When sketching, it is helpful to keep in mind that even functions are symmetric about the y-axis and that odd functions are symmetric about the origin (0, 0).

9. Functions and Absolute Value Even Function Odd Function

10. Functions and Absolute Value ABSOLUTE VALUE

11. Functions and Absolute Value • Definition:

12. Functions and Absolute Value Example. Given that show that

13. Functions and Absolute Value SKETCHING – THE USE OF CASES

14. Functions and Absolute Value • How to sketch functions involving piecewise definitions? • Start by looking for the key x-values where the function changes value • Use these x-values to create different “cases” • Recall: (Heaviside function)

15. Functions and Absolute Value

16. Functions and Absolute Value Example. Sketch • Therefore, key points are x = -1 and x = 0 • 342

17. Functions and Absolute Value Example. Sketch • Cases: • In case 1, we have . • In case 2, we have . • In case 3, we have

18. Functions and Absolute Value

19. Functions and Absolute Value

20. Functions and Absolute Value Example. Sketch the inequality . • Case 1: , which implies that • We have • Isolating for : • Case 2: , which implies that • We have • Isolating for :

21. Functions and Absolute Value

22. Functions and Absolute Value

23. One-to-One Functions & Inverses ONE-TO-ONE FUNCTIONS

24. Functions and Absolute Value • A function is one-to-one if it never takes the same y-value twice, that is, it has the property: • Horizontal Line Test: We can see that a function is one-to-one if any horizontal line touches the function at most once. • If a function is increasing and decreasing on different intervals, it cannot be one-to-one unless it is discontinuous.

25. One-to-One Functions & Inverses

26. One-to-One Functions & Inverses y = ln(x) y = cos(x)

27. One-to-One Functions & Inverses

28. Functions and Absolute Value

29. One-to-One Functions & Inverses INVERSE FUNCTIONS

30. One-to-One Functions & Inverses • A function that is one-to-one with domain A and range B has an inverse function with domain B and range A. • reverses the operations of in the opposite direction • is a reflection of in the line y = x

31. One-to-One Functions & Inverses Cancellation Identity:Let and be functions that are inverses of each other. Then: The cancellation identity can be applied only if x is in the domain of the inside function.

32. One-to-One Functions & Inverses

33. One-to-One Functions & Inverses 11

34. One-to-One Functions & Inverses INVERSE TRIGONOMETRIC FUNCTIONS

35. One-to-One Functions & Inverses In order to define an inverse trigonometric function, we must restrict the domain of the corresponding trigonometric function to make it one-to-one.

36. One-to-One Functions & Inverses

37. One-to-One Functions & Inverses rgregr

38. One-to-One Functions & Inverses • Example. Simplify . Let . Then, . Constructing a diagram: By Pythagorean Theorem, missing side has length Thus, egegge

39. Limits EVALUATING LIMITS

40. Limits Limit Laws Given the limits exist, we have:

41. Limits Advanced Limit Laws Given the limits exist and n is a positive integer, we have: • Indeterminate Form (can’t use limit laws) • You must algebraically work with the function (by factoring, rationalizing, and/or expanding) in order to get it into a form where the limit can be determined.

42. Limits Example. Evaluate 111

43. Limits Example. Evaluate 111

44. Limits THE FORMAL DEFINITION OF A LIMIT

45. Limits if given any , we can find a such that:

46. Limits Set } Select

47. Limits SQUEEZE THEOREM

48. Limits Squeeze Theorem: and then

49. Limits ----

50. Limits Fundamental Trigonometric Limit: