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# Unit 1 limits and continuity

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### Unit 1 limits and continuity

1. 1. Objective: SWBAT examine multiple representations ofa function in order to become familiar with propertiesof common functions0011 0010 1010 1101 0001 0100 1011 DRILL: August 30, 2011 My Expectations of Calculus…0011 0010 1010 1101 0001 0100 1011 •Why did you sign up for calculus? 1 2 4 •What do you expect the year to be like? •What are your plans after high school that involve mathematics or science-related fields?
2. 2. 0011 0010 1010 1101 0001 0100 1011 1 2 4 This lab assignment can be found on EDLINE.
3. 3. The Dominance of Functions • Any exponential function of n dominates any0011 0010 1010 1101 0001 0100 1011 polynomial function of n. • Any polynomial function of n dominates and logarithmic function of n. • Any logarithmic function of n dominates a constant term. 1 2 • Any polynomial of degree k dominates a polynomial of degree l if and only if k>l 4 In general, x(n) dominates y(n) if and only if grows large. grows as n
4. 4. Exit Ticket0011 0010 1010 1101 0001 0100 1011 1 2 4
5. 5. Homework0011 0010 1010 1101 0001 0100 1011 Mathematical Autobiography • Typed • Double-spaced • Times Roman, 12 Font 1 2 4 • 4 paragraphs (as outlined on worksheet) (one paragraph was done as today’s drill)
6. 6. Objective: SWBAT use area representations in order toevaluate limits using tables, graphs, and functions0011 0010 1010 1101 0001 0100 1011 DRILL QUIZ #1: September 1, 20110011 0010 1010 1101 0001 0100 1011 1 2 4
7. 7. 0011 0010 1010 1101 0001 0100 1011 1 2 4 This lab assignment can be found on EDLINE.
8. 8. Evaluating Limits of Functions TABLES0011 0010 1010 1101 0001 0100 1011 GRAPHS 1 2 SUBSTITUTION 4
9. 9. Limit Existence from a Graph Conclusion:0011 0010 1010 1101 0001 0100 1011 Existence or nonexistence at x=c has no effect on the existence of the limit of the 2 function at x=c. 1 4
10. 10. Limit Non-Existence from a Graph Conclusion:0011 0010 1010 1101 0001 0100 1011 The limit at x=c does not exist if the function has oscillating or unbounded behavior or a 2 jump discontinuity at x=c. 1 4
11. 11. Exit Ticket0011 0010 1010 1101 0001 0100 1011 1 2 4
12. 12. Homework0011 0010 1010 1101 0001 0100 1011 Calculus Textbook • Pgs. 54-58 • #’s 2, 7, 8, 9-18, 20, 26, 60, 63, 65, 66 1 2 4 • Pg. 67 • Choose one from #’s 11-22
13. 13. Objective: SWBAT use direct substitution and otheralgebraic manipulations in order to evaluate limits0011 0010 1010 1101 0001 0100 1011 DRILL QUIZ #2: September 6, 20110011 0010 1010 1101 0001 0100 1011 1 2 4
14. 14. The Indeterminate Form • This limit cannot be determined0011 0010 1010 1101 0001 0100 1011 • But this does not mean that the limit DNE 1 2 When this happens, try the following: • Factor 4 • Rationalize the numerator or denominator • Use Trig Substitutions to rewrite the function
15. 15. Exit Ticket0011 0010 1010 1101 0001 0100 1011 Evaluating Limits Worksheet 1 2 4
16. 16. Homework0011 0010 1010 1101 0001 0100 1011 Calculus Textbook • Pgs. 67-69 • #’s 24, 27, 35, 42, 44, 52, 54, 65, 67, 70, 77, 97, 98 1 2 4 • Prove the 2nd Special Trig Limit involving cosine
17. 17. Objective: SWBAT examine the area of regular polygons in order to evaluate limits at infinity DRILL QUIZ #3: September 8, 20110011 0010 1010 1101 0001 0100 10110011 0010 1010 1101 0001 0100 1011 1 2 4
18. 18. 0011 0010 1010 1101 0001 0100 1011 1 2 4 This lab assignment can be found on EDLINE.
19. 19. Horizontal Asymptote0011 0010 1010 1101 0001 a horizontal The line y = L is 0100 1011 asymptote of the graph of f if or 1 2 4
20. 20. Limits at Infinity of Rational Functions • If the degree of the numerator is less than the0011 0010 1010 1101 0001 0100 1011 degree of the denominator, then the limit of the rational function is 0. • If the degree of the numerator is equal to the 1 degree of the denominator, then the limit of 2 the rational function is the ratio of the leading coefficients. 4 • If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.
21. 21. Exit Ticket0011 0010 1010 1101 0001 0100 1011 1 2 4
22. 22. Homework0011 0010 1010 1101 0001 0100 1011 Calculus Textbook • Pgs. 205-207 • #’s 1-8, 15-20, 88b 1 2 4