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The deriva*ve
Geometric view
UBC Math 102
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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://www.math.ubc.ca/~MLC/.
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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.
UBC Math 102
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Course Calendar:
You are here
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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
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Prac*ce calcula*on we did in class
UBC Math 102
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Help with homework
UBC Math 102
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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
UBC Math 102
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1. Assignment2: Problem 16
• The limit is (A) 1 (B) 2/3 (C) 5/4 (D) 9/10 (E) 0
UBC Math 102
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Assignment2: Problem 16
• Soln:
• Now evaluate at x=7 to get 9/10
UBC Math 102
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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
UBC Math 102
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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
UBC Math 102
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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
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Defn: the deriva*ve
• The deriva*ve of a func*on at is
• We denote this by the nota*on or
UBC Math 102
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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.
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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
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Desmos: tangent line • hOps://www.desmos.com/calculator/fs3lx8gptb
• Experiment with slider • Experiment with zoom!
UBC Math 102
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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
UBC Math 102
o
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Zooming into the graph of
• At the point x=1.5
UBC Math 102 See course notes
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Zooming into the graph of
At x=0
UBC Math 102 See course notes
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A func*on with a cusp
At the cusp, the deriva*ve does not exist.
UBC Math 102 See course notes
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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.
UBC Math 102
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My week at Banff Interna*onal Research Sta*on (BIRS)
September 2014
UBC Math 102
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My former students
UBC Math 102
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Banff winter in September!
UBC Math 102
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Outdoor fun
UBC Math 102
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Spectacular Rockies!
UBC Math 102
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UBC Math 102
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UBC Math 102
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UBC Math 102
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UBC Math 102
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UBC Math 102
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UBC Math 102
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3. Where is the slope greatest?
UBC Math 102
A B C
D E
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4. Where is the slope zero?
UBC Math 102
A B C
D E
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5. Is that the only place? (is there another point where the slope is
zero?) • A: no • B: yes
UBC Math 102
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6. Where is the slope nega*ve?
UBC Math 102
A B C
D E
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Sketch the deriva*ve of this func*on
UBC Math 102
f(x)
x
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7. What does your sketch look like?
UBC Math 102
f(x)
x A B C D
f ’(x)
x
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8. What is this func*on?
• A |x+1| • B |x-‐1| • C -‐|x+1| • D -‐|x-‐1| • E -‐ (x-‐1)
UBC Math 102
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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
UBC Math 102
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Sketch the deriva*ve of this func*on
UBC Math 102
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10. The deriva*ve of this func*on looks like:
• A
• B
UBC Math 102
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My drive from Banff to Calgary
• S
• Shown is the distance (S) versus *me (t)
UBC Math 102
t
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11. My drive from Banff to Calgary
• S
• At what *me were we moving fastest?
UBC Math 102
t A B C D E
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12. My drive from Banff to Calgary
• S
• When did we get stuck in Calgary traffic?
UBC Math 102
t A B C D E
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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
UBC Math 102
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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
UBC Math 102
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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
UBC Math 102
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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)
UBC Math 102
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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
UBC Math 102
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Solu*on to problem from last *me
UBC Math 102
SOLUTION ON NEXT SLIDE à
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