Today... Midterm sign-up form available after add/drop date. OSH submission - look for Crowdmark 2 days before due date.

Transcription

1 Today... Midterm sign-up form available after add/drop date. OSH submission - look for Crowdmark 2 days before due date. Comment on Diagnostic Test. Asymptotics (approximations when x is small or large) Reminder: CL due Monday 7am. Reminder: PL2.1 due Tuesday 7am. Reminder: OSH 0 due Wednesday 7am. Reminder: PL2.2 due Thursday 7am. Reminder: OSH 1 due Friday 8:59 pm. Reminder: Diagnostic due Sunday 11:59pm (Sept 17).

2 Diagnostic

3 Diagnostic Treat it like a test. Take about an hour.

4 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong).

5 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong). Review HS material BEFORE if you want but do it without looking at notes, googling, talking to friends.

6 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong). Review HS material BEFORE if you want but do it without looking at notes, googling, talking to friends. You ll get advice at the end on what you need to work on.

7 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong). Review HS material BEFORE if you want but do it without looking at notes, googling, talking to friends. You ll get advice at the end on what you need to work on. FOLLOW THE ADVICE!

8 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong). Review HS material BEFORE if you want but do it without looking at notes, googling, talking to friends. You ll get advice at the end on what you need to work on. FOLLOW THE ADVICE! Holes in background are BAD.

9 Diagnostic Treat it like a test. Take about an hour. You get points for submitting each answer (right or wrong). Review HS material BEFORE if you want but do it without looking at notes, googling, talking to friends. You ll get advice at the end on what you need to work on. FOLLOW THE ADVICE! Holes in background are BAD. Link to review material at the end - not for points (not directly, at least).

10 Studying experiment On a piece of paper, write down your name (first and last), date of birth (lie if you don t trust strangers), location of birth, your intended major, your career ambitions (dream big!), a list of places you ve lived, 3 favourite subjects from high school, your first pet s name or an instrument you play(ed), your phone number (again, lie if necessary). Trade with someone you don t know sitting near you. Read their info and try to remember it. Give the paper back to your neighbour.

11 Asymptotics - approximations when x is small or large

12 Asymptotics - approximations when x is small or large is only small compared to something like 1.

13 Asymptotics - approximations when x is small or large is only small compared to something like 1. Compared to , it s big.

14 Asymptotics - approximations when x is small or large is only small compared to something like 1. Compared to , it s big. Small and big are relative.

15 Asymptotics - approximations when x is small or large is only small compared to something like 1. Compared to , it s big. Small and big are relative. It s only safe to ignore something small when it s being added to something big, not multiplied.

16 Asymptotics - approximations when x is small or large is only small compared to something like 1. Compared to , it s big. Small and big are relative. It s only safe to ignore something small when it s being added to something big, not multiplied. Sometimes use notation << 1.

17 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0).

18 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). If you make the wall 20 times as high, it is still small (20a 0).

19 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). If you make the wall 20 times as high, it is still small (20a 0). Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

20 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

21 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

22 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). Depends. Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

23 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). Depends. Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

24 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). Depends. Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

25 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). True. Depends. Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall.

26 Comparisons and approximation must be based on relative sizes! For each of the following, (A) True, (B) False, (C) Not sure... You line up some bricks to make a wall one brick high. The wall is small (a 0). Depends. If you make the wall 20 times as high, it is still small (20a 0). True. Depends. Add one row of bricks to a wall 20 bricks high. The new wall is about the same size as the old wall. a+b b

27 Back to the experiment Left side of room - show your piece of paper to your neighbour and let them read over it again. Right side of room - do not show your neighbour the paper again but ask them to repeat the info as best they can. After they do so, show them the paper.

28 When x << 2, then x + 2 can be approximated by... A. 2 B. x C. infinity D. Don t know - please explain. (Assume x>0.)

29 When x << 2, then x + 2 can be approximated by... A. 2 B. x C. infinity D. Don t know - please explain. (Assume x>0.)

30 Back to the experiment Left side of room - show your piece of paper to your neighbour and let them read over it again. Right side of room - do not show your neighbour the paper again but ask them to repeat the info as best they can. After they do so, show them the paper.

31 When x >> b, then x + b can be approximated by... A. b B. x C. infinity D. Don t know - please explain. (Assume x, b are positive.)

32 When x >> b, then x + b can be approximated by... A. b B. x C. infinity D. Don t know - please explain. (Assume x, b are positive.)

33 When x << 1, then x 2 - x 3 can be approximated by... A. x 2 B. -x 3 C. None of the above. D. Don t know - please explain.

34 When x << 1, then x 2 - x 3 can be approximated by... A. x 2 B. -x 3 C. None of the above. D. Don t know - please explain.

35 Back to the experiment Left side of room - show your piece of paper to your neighbour and let them read over it again. Right side of room - do not show your neighbour the paper again but ask them to repeat the info as best they can. After they do so, show them the paper.

36 Which of the following provides a good approximation to the graph of f(x)= x 2 - x 3 near the origin? (A) (B) (C) (D)

37 Which of the following provides a good approximation to the graph of f(x)= x 2 - x 3 near the origin? (A) (B) (C) (D)

38 Which of the following provides a good approximation to the graph of f(x)= x 2 - x 3 near the origin? x 2 (A) (B) x 2 - x 3 (C) (D)

39 Which of the following provides a good approximation to the graph of f(x)= x 2 - x 3 near the origin? x 2 (A) (B) x 2 - x 3 (C) (D)

40 When x >> 1, then x 2 - x 3 can be approximated by... A. x 2 B. -x 3 C. None of the above. D. Don t know - please explain.

41 When x >> 1, then x 2 - x 3 can be approximated by... (A) x 2 (B) -x 3 (C) None of the above. (D) Don t know - please explain.

42 When x >> 1, then x 2 - x 3 can be approximated by... (A) x 2 (B) -x 3 x 2 - x 3 (C) None of the above. (D) Don t know - please explain. - x 3

43 When x >> 1, then x 2 - x 3 can be approximated by... (A) x 2 (B) -x 3 x 2 - x 3 (C) None of the above. (D) Don t know - please explain. - x 3

44 When x >> 1, then x 2 - x 3 can be approximated by... (A) x 2 (B) -x 3 x 2 - x 3 (C) None of the above. (D) Don t know - please explain. - x 3

45 Back to the experiment Both sides of room - do not show your neighbour the paper again but ask them to repeat the info as best they can. After they do so, show them the paper.

46 Back to the experiment Both sides of room - do not show your neighbour the paper again but ask them to repeat the info as best they can. After they do so, show them the paper. (A) All 9 pieces of information correct. (B) 8 pieces of information correct. (C) 7 pieces of information correct. (D) 5-6 pieces of information correct. (E) 0-4 pieces of information correct.

47 Back to the experiment

48 Back to the experiment Retrieval is critical for memory/learning.

49 Back to the experiment Retrieval is critical for memory/learning. Reading notes or watching a lecture not as good as actively accessing.

50 Back to the experiment Retrieval is critical for memory/learning. Reading notes or watching a lecture not as good as actively accessing. Interleaving versus blocking.

51 Back to the experiment Retrieval is critical for memory/learning. Reading notes or watching a lecture not as good as actively accessing. Interleaving versus blocking. Desirable difficulties improve long term recall.

52 Back to the experiment Retrieval is critical for memory/learning. Reading notes or watching a lecture not as good as actively accessing. Interleaving versus blocking. Desirable difficulties improve long term recall. If a method of learning feels easier - be skeptical that it s better!

53 Back to the experiment Retrieval is critical for memory/learning. Reading notes or watching a lecture not as good as actively accessing. Interleaving versus blocking. Desirable difficulties improve long term recall. If a method of learning feels easier - be skeptical that it s better! Math is not a spectator sport! (any learning, really)

54 Coming up next lecture Hill functions - a family of rational functions that comes up in many models in biology. Using asymptotics to understand their shape. Sketching them.