the$derivave$ - math.ubc.cakeshet/m102/2015/lect3.1.pdf · math$learning$center$ •...

The deriva*ve

Geometric view

UBC Math 102

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•  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://www.math.ubc.ca/~MLC/.

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.

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Course Calendar:

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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

Prac*ce calcula*on we did in class

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Help with homework

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Assignment2: Problem 16

•  Evaluate the limit, if it exists. If a limit does not exist, enter DNE.

•  Hint: consider factoring top and boOom

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1. Assignment2: Problem 16

•  The limit is (A) 1 (B) 2/3 (C) 5/4 (D) 9/10 (E) 0

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•  Soln:

•  Now evaluate at x=7 to get 9/10

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•  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

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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

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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 xàa, “the limit DNE” UBC Math 102

Defn: the deriva*ve

•  The deriva*ve of a func*on at is

•  We denote this by the nota*on or

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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.

Desmos: Secant line •  hOps://www.desmos.com/calculator/mekjo0w38

•  Slope:

•  Experiment with changing a and h (use slidersà)

We can see this even

beOer using

Desmos: tangent line •  hOps://www.desmos.com/calculator/fs3lx8gptb

•  Experiment with slider •  Experiment with zoom!

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Tangent line

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

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o

Zooming into the graph of

•  At the point x=1.5

UBC Math 102 See course notes

Zooming into the graph of

At x=0


A func*on with a cusp

At the cusp, the deriva*ve does not exist.


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.

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

September 2014

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My former students

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Banff winter in September!

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Outdoor fun

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Spectacular Rockies!

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3. Where is the slope greatest?

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A B C

D E

4. Where is the slope zero?

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A B C

D E

5. Is that the only place? (is there another point where the slope is

zero?) •  A: no •  B: yes

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6. Where is the slope nega*ve?

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A B C

D E

Sketch the deriva*ve of this func*on

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f(x)

x

7. What does your sketch look like?

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f(x)

x A B C D

f ’(x)

x

8. What is this func*on?

•  A |x+1| •  B |x-‐1| •  C -‐|x+1| •  D -‐|x-‐1| •  E -‐ (x-‐1)

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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

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Sketch the deriva*ve of this func*on

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

•  A

•  B

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My drive from Banff to Calgary

•  S

•  Shown is the distance (S) versus *me (t)

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t

11. My drive from Banff to Calgary

•  S

•  At what *me were we moving fastest?

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t A B C D E

12. My drive from Banff to Calgary

•  S

•  When did we get stuck in Calgary traffic?

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t A B C D E

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

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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

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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

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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)

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Related test-‐type problem

Use the defini*on of the deriva*ve to compute the deriva*ve of the above func*on

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Solu*on to problem from last *me

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SOLUTION ON NEXT SLIDE à

the$derivave$ - math.ubc.cakeshet/m102/2015/lect3.1.pdf · math$learning$center$ •...

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