The deriva*ve. Geometric view. UBC Math 102

Transcription

1 The deriva*ve Geometric view

2 Math Learning Center The MLC is a space for undergraduate students to study math together, with friendly support from tutors (grad students in math). Located at LSK310 and LSK302 Hours: open 5 days a week. Note: MLC is not a place to check HW answers or receive solu*ons. MLC aim is to aid students in becoming beoer learners and to develop cri*cal thinking skills. hop://

3 Announcements If you have been aoending but not yet registered in this sec*on, please see me ater class. There is a quiz this Friday (Sept 25). It will involve both individual and group work.

4 Course Calendar: You are here

5 Office hrs: Regular office hrs: Mon 4-5pm Wed 3-4pm Thurs 12:30-1:30pm* (*except on days when I have a dept mee2ng at that 2me) Math Annex 1111

6 Prac*ce calcula*on we did in class

7 Help with homework

8 Assignment2: Problem 16 Evaluate the limit, if it exists. If a limit does not exist, enter DNE. Hint: consider factoring top and booom

9 1. Assignment2: Problem 16 The limit is (A) 1 (B) 2/3 (C) 5/4 (D) 9/10 (E) 0

10 Assignment2: Problem 16 Soln: Now evaluate at x=7 to get 9/10

11 Assignment2: Problem 18 Evaluate the limit by guessing We have not learned enough about this func*on to do this, but we can use a value close to x=0 to guess the limit

12 2. Assignment2: Problem 18 To guess a value for this limit: I would input which of the following into WebWork? (A) (B) (C) (D) (E) None of the above

13 Prac*cal guide to limits To evaluate Factor f(x) and simplify, then plug in x=0 or Find the value of f(x) close to x=a. If f(x) approaches the same value, for xà a+ (slightly above) and xà a- (slightly below) x=a, that value is the limit. If f(x) approaches two different values from above and below x=a, then the limits from above and from below are not the same. In that case, we say THE limit DNE If values of f(x) get larger and larger as xà a, the limit DNE

14 Defn: the deriva*ve The deriva*ve of a func*on at is We denote this by the nota*on or

15 Geometric point of view Now we switch gears and talk about secant lines, tangent lines, deriva*ves and all that from the point of view of graphs and pictures.

16 We can see this even beoer using Desmos: Secant line hops:// Slope: Experiment with changing a and h (use slidersà )

17 Desmos: tangent line hops:// Experiment with slider Experiment with zoom!

18 Tangent line It is the line we see when we zoom into the graph of a func*on at some point o

19 Zooming into the graph of At the point x=1.5 See course notes

20 Zooming into the graph of At x=0 See course notes

21 A func*on with a cusp At the cusp, the deriva*ve does not exist. See course notes

22 When does the deriva*ve exist? If a func*on is discon*nuous at x=a then its deriva*ve is not defined at that point. If a func*on has a cusp at x=a, then its deriva*ve does not exist (is not defined) at that point.

23 My week at Banff Interna*onal Research Sta*on (BIRS) September 2014

24 My former students

25 Banff winter in September!

26 Outdoor fun

27 Spectacular Rockies!

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34 3. Where is the slope greatest? D E A B C

35 4. Where is the slope zero? D E A B C

36 5. Is that the only place? (is there another point where the slope is A: no B: yes zero?)

37 6. Where is the slope nega*ve? D E A B C

38 Sketch the deriva*ve of this func*on f(x) x

39 7. What does your sketch look like? f(x) f (x) A B C D x x

40 8. What is this func*on? A x+1 B x- 1 C - x+1 D - x- 1 E - (x- 1)

41 9. This func*on is A discon*nuous B not differen*able anywhere C has no limit as x- >0 D differen*able except at x=1 E I am confused

42 Sketch the deriva*ve of this func*on

43 A 10. The deriva*ve of this func*on looks like: B

44 My drive from Banff to Calgary S Shown is the distance (S) versus *me (t) t

45 11. My drive from Banff to Calgary S A B C D E At what *me were we moving fastest? t

46 12. My drive from Banff to Calgary S A B C D E When did we get stuck in Calgary traffic? t

47 Graphing ac*vity LG (1) to be able to construct rough sketches of common func*ons and their rela*ves LG (2) given a func*on, to be able to sketch the deriva*ve of that func*on

48 This is sin(x) Graph f(x)= sin(x) Graph the deriva*ve g(x)=f (x) Find the equa*on of the tangent line at x=a Draw the tangent line on your graph

49 Answers 1 D 2 D 3 D 4 A 5 B 6 B 7 C 8 D 9 D 10 A 11 E 12 C or D

50 Related test problem Shown in the graph is the velocity of a par*cle. Use this to sketch the accelera*on of the par*cle. (Hint: the accelera*on is the deriva*ve of the velocity)

51 Related test- type problem Use the defini*on of the deriva*ve to compute the deriva*ve of the above func*on

52 Solu*on to problem from last *me SOLUTION ON NEXT SLIDE à

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