MATH 127 MIDTERM 1

1. MATH 127 MIDTERM 1

2. Tutor: Maysum Panju • 3B Computational Mathematics • Lots of tutoring experience • Interests: • Reading • Pokémon • Calculus

3. 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

4. Outline • Introduction • Sets, inequalities, absolute values • Basic function concepts • Exponentials and Logarithms • Trigonometry • Limits • Second Degree Equations • Questions

5. Writing Solutions A good solution includes… • An introductory statement: what you are given and what you have to show/find; • A concluding statement: summarize the conclusion briefly; • Justifications of the main steps: refer to definitions, rules, and known properties; • Some sentences of guidance for the reader, e.g. how you are going to solve the problem.

6. Sets of Numbers • Set builder notation:S = { x | x satisfies some property } • Sets you should know: • Intervals:

7. Absolute Values • Absolute value: the “size” of a number

8. Absolute Value Inequality

9. Absolute Value Equation Solve for x:

10. Introduction to Functions

11. What is a Function? • Function: Turns objects from one set into objects in another set. • For each x in X, assign some value y in Y. • You can only assign one y per x. • This implies the “Vertical Line Test”! X Y f

12. Domain and Range • Domain: What is the input (x) allowed to be? • Range: What values (y) does the function hit?

13. Even and Odd Functions • Even functions: • Reflect along y-axis • e.g. Polynomials with only even degree terms • Odd functions: • Reflect around the origin • e.g. Polynomials with only odd degree terms

14. Even and Odd Polynomials

15. Increasing /Decreasing Functions • Increasing functions: implies • As x increases, so does f(x). • Odd functions: implies • As y increases, so does f(y). • Note: these inequalities are “strict”.

16. Transformation of Functions • Given a function We can transform it: • Here, • Compress f horizontally by k (reflect in y-axis if k < 0) • Translate f to the right p units • Stretch f vertically by a (reflect in x-axis if a < 0) • Translate f upwards by q • Horizontal transformations are “backwards” and appear inside the function f. These affect the domain. • Vertical transformations are “normal” and appear outside the function f. These affect the range.

17. Example

18. Function Compositions • Given two functions f and g, sometimes we can “compose” them to make a new function. • Given and The composition is given as where C A B f g

19. Composition of Functions • You can only compose functions when the range of the innerfunction is within the domain of the outer function • If the inner function hits a value that the outer function isn’t defined on, there’s a problem!

20. Example Domain of Domain of

21. Graphing Reciprocals • Given a function , graphing the reciprocal is easy: • Find all points where These points remain fixed. • Find all points where These become vertical asymptotes. • Increasing sections become decreasing sections, and vice versa.Positive sections remain positive sections, and vice versa.Maxima become minima, and vice versa. • Check any special points on a point-wise basis.

22. Example Given , graph .

23. Inverse Functions • An inverse function: Given a function f, define a new function f -1that “undoes” f. • The inverse must be a function: • Must pass the vertical line test! • A function has a inverse if and only if • It is “one-to-one” (passes the horizontal line test) • It is strictly increasing or decreasing (if continuous) • Do NOT confuse f -1with 1/f !!

24. Inverse Functions • If we have a functionThen the inverse function satisfies • Sometimes we must restrict the domain of f to ensure that the inverse is a function. • In this case, the range of the inverse f -1is restricted.

25. Finding inverse functions • If you have an equation , swap x with y and rearrange for y. • If you have to choose between +/-, take the +. This corresponds to restricting the domain. • If you have a graph, reflect the graph in the line y = x. • Remember to restrict the domain to a 1-to-1 interval first (so it passes the horizontal line test)! • Example: find the inverse ofon a suitable interval.

26. Exponents and Logarithms

27. Exponent Laws • Some common exponent laws: • In general, the exponential operation is really powerful. • Weak operations in exponents become stronger once you pull them out. • Examples: • Addition in the exponent becomes multiplication outside. • Multiplication in the exponent becomes exponentiation outside.

28. Exponential Graphs • Given , the shape of the graph depends entirely on the choice of a. If a > 1 If 0 < a < 1

29. Logarithm Laws • Think of logs as the inverse of exponentiations. • In general, the logarithm operation is really weak. • Strong operations in logarithms become weaker once you pull them out. • Examples: • Multiplication in the log becomes addition outside. • Exponents in the log become multiplication outside.

30. Logarithm Graphs • Given , the shape of the graph depends entirely on the choice of a. If a > 1 If 0 < a < 1

31. Example problem • Graph the function • What is the domain? • What is the range? • Find the equation of the inverse on a suitable interval.

32. Example - Solution • Graph: • Domain: • Range: • Inverse on : (0, 2)

33. Break Time!

34. Trigonometry

35. Review of Trigonometry • Main ratios: sin, cos, tan (SOHCAHTOA) • Reciprocals: csc, sec, cot Unit Circle: r = 1

36. TrigGraphs http://www.algebra-help.org/graphs-of-trigonometric-functions.html

37. Trig Graphs http://www.eighty-twenty.org/

38. Trig Equations: Formulas to Know • Compound Angles:

39. Example Problem • Compute . • Solution:

40. Inverse Trig FUNCTIONS arcsin arccos arctan

41. Examples Prove that . Picture proof…

42. Limits and Tangents

43. Limits • In calculus, the main ideas involve working with very small numbers and very big numbers. • Limits help us… • Use extremely small values • Reach really large values • Predict the value of a function at a place it isn’t defined • Relax the rules of domain restrictions • Heuristically: the limit of a function at a point is the value the function “should” take at that point if it were nice and smooth.

44. Left and Right Limits • The left limit of f(x) at the point x = a is the value the function approaches as x gets close to a from the left. • The right limit of f(x) at the point x = a is the value the function approaches as x gets close to a from the right.

45. Limits • If the left limit of f(x) AND the right limit of f(x) BOTH exist at the point x = a, and are EQUAL, then we say the limit of f(x) at a exists as well. • In this case, the limit is equal to the limits from the left and right. so

46. How to think about Limits • When considering the limit of f(x) as x approaches a … Think of the curve here and here But NOT here

47. Computing a Limit • Computing limits is an art. • Adapt the technique for the problem! • Given a graph, guess the limit by inspection. No Limit Limit = L

48. Computing a Limit • Given • First thing to try: Substitute x = a in the equation. If it works, and the function is “continuous”, you’re done! • Usually (in “interesting” problems), you get an indeterminate form:

49. Computing a Limit • Some tricks to try: • Factor and cancel out the problems. • Multiply by A/A for some clever A to eliminate square roots (set up a difference of squares) • Use trig identities to help you cancel things • If absolute values are involved, use cases (show the limit does NOT exist by showing left limit is not the same as right limit) • Change the variables • Squeeze Law

50. Change of Variables • To turn a limit to a more recognizable form, try a variable change. • Example: • If x approaches infinity, let h = 1/x. Then h approaches 0. • Try to get to familiar limits you can apply. • Practice: