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Linear Approxima-on
and Newton’s method for finding zeros
UBC Math 102
Announcements
• Sign up for Midterm: • Via the Wiki or this link:
hHp://www.math.ubc.ca/~cytryn/teaching/math102F15/MATH102SignupSheet.php
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Course Calendar:
•
You are here
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
Calcula-on we did in class (1)
Find the deriva-ve of Solu-on: by the power rule:
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Calcula-on we did in class (2)
•
• (Find x1)
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Solu-on (in class) to (2)
Eqn of tangent line at x0 (from last -me)
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Microscopic mo-on: The inner life of the cell
• Starring role: kinesin • www.youtube.com/watch?v=wJyUtbn0O5Y
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Fantas-c vesicle traffic • www.youtube.com/watch?v=7sRZy9PgPvg
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Motion of Molecular motors carrying cargo inside a long (fungus) cell.
Martin Schuster et al. PNAS 2011;108:3618-3623
©2011 by National Academy of Sciences
Vesicle mo-on: vesicles carried by molecular motors along microtubule
tracks
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distance
-me
Simplified view
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Simplified vesicle mo-on
• Vesicle 1: Sketch the velocity and describe what was happening to the vesicle during this -me.
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• Vesicle 2: Sketch the displacement and describe what was happening to the vesicle during this -me.
Linear approxima-on
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(1) Tangent line equa-on
Which of the following is the equa-on of the tangent line to f(x) at the point x0
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(2) Linear approxima-on Close to the point x0 we can
approximate the func-on f(x) by
(3) Linear approxima-on
The approxima-on is good provided • (A) x is small • (B) x0 is small • (C) (x-‐x0) is small • (D) (x0-‐x) is large • (E) None of the above
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(4) A linear approxima-on..
represents the true value of the func-on at some point close to x0 by the value shown in
• A, B, C, D, or E UBC Math 102
A
B
C
D E
Geometry
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f(x0)
Δf
Geometry
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f(x0)
(x-‐x0)
f’(x0)(x-‐x0)
Δf
Δf
(5) Example:
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(6) Example cont’d
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(7) Example cont’d
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(8) Over or under-‐es-mate? The approxima-on we made will (A) Over-‐es-mate the true value (B) Under-‐es-mate “ “ “
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PART 1
• A demo of linear approxima-on on Desmos
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Desmos
Graph the func-on: Use desmos to graph its deriva-ve
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It should look like:
•
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Desmos cont’d
Add the equa-on of a generic tangent line at x=a A slider should appear for a so you can shin the loca-on of that tangent line. Add the point (a, f(a)) to your graph.
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It should look like:
•
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Using Desmos
Consider the tangent line at xo=5. Use your graph to find the values below: f(5), f(6) and the linear approx to f(6)
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(9) Using Desmos
I got the values. f(5), f(6) and the linear approx to f(6) (A) 4.37, 4, 5 (B) 4.37, 5, 2 (C) 4.37, 2, 2.5 (D) huh??
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PART 2
• A demo of Newton’s Method on Desmos
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Intro to Newton’s Method
We will use the same example to mo-vate another use of the tangent line
FINDING ZEROS OF A FUNCTION
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Find zero of f and find where the tangent line intersects x axis
(Use the tangent line at x0=5).
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(10) Find zero of f and find the value where the tangent line intersects x
axis (Use the tangent line at x0=5). (A) The zero of f is at 6, the TL “zero” is at 8 (B) The zero of f is at 7, the TL “zero” is at 8 (C) The zero of f is at 6.3, the TL “zero” is at 7.1 (D) The zero of f is at 6.587, TL “zero” is at 7.458 (E) The zero of f is at 6.855, the TL “zero” is at
7.55
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You can click on the graph to find the coordinates
• TL and f(x)
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Desmos does the work
Get Desmos to find the point at which the tangent line crosses the x axis: • Use our result from the intro to this lecture! • Add the “formula” for x1 to your Desmos graph.
• Add the point (x1,0) to your graph
Should look like:
•
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Now we will repeat the process
To get closer to the zero of f: • Construct a tangent line at the point x1 • Find the point where the new tangent line intersects x axis (Find x2).
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Should look like:
•
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To No-ce:
• We are geqng closer to the actual zero of the func-on.
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Effect of ini-al guess
Go to the slider for x0 and shin it to the value 3. What happens to the tangent lines? What happens to the approxima-on for the zero of the func-on?
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Answers
• 1 C (Need to use both f and f’ at x_0) • 2 E (uses eqn of tangent line for the approx) • 3 C • 4 B (uses pt on tangt line as approx near x_0) • 5 D • 6 B (power rule) • 7 A • 8 A • 9 C • 10 D
Related test-‐type problem:
Find a linear approxima-on that provides a rough es-mate for the value of (1.1)8. Explain why the approxima-on is an (over/under)-‐es-mate.
• (Note: a calculator value is 2.1435)
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Related test ques-on (MT1, 2014) (Tangent lines)
Consider the func-on (a) At which points (a,f(a)) does the graph of this
func-on have tangent lines parallel to the line y = −x.
(b) What is the equa-on of the tangent lines at each of these points.
solu-on
Solu-on:
(a)
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Solu-on
(b)
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Test ques-on (Quo-ent rule, ra-onal func-ons) MT1 2014
At an all-‐you-‐can-‐eat buffet, the total calories you gain can be represented by the func-on where t ≥ 0 is the -me in minutes you spend at the restaurant and A and b are posi-ve constants. • (a) If you stayed for a long -me, what asymptote would your total caloric gain approach?
• (b) Aner how much -me do you gain exactly half of that asympto-c caloric amount?
• (c) At -me t, what is the instantaneous rate at which your caloric gain changes? solu-on
Solu-on:
(a) (b) (c)
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