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Summary of Outreach Trip to Cusco, Peru – Building Educational Futures

From August 20 to September 4, 2010, a group of 22 students participated in an outreach trip to Cusco, Peru, with a project cost of $16,000. The focus of the trip was on building education infrastructure, including constructing a kindergarten classroom to provide free education, as well as a sewing workshop to improve job prospects for local women. An English Language Teaching (ELT) classroom was also established to further enhance the employment opportunities for community members. For more information, visit studentsofferingsupport.ca/blog.

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Summary of Outreach Trip to Cusco, Peru – Building Educational Futures

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  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:

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