Download - Lesson 1: The Tangent and Velocity Problems
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Section 2.1The Tangent and Velocity Problems
Math 1a
February 1, 2008
Announcements
I Grab a bingo card and start playing!
I Syllabus available on course websiteI Homework for Monday 2/4:
I Practice 2.1: 1, 3, 5, 7, 9I Turn-in 2.1: 2, 4, 6, 8I Complete the ALEKS initial assessment (course code
QAQRC-EQJA6)
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Outline
Bingo
Velocity
Tangents
![Page 3: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/3.jpg)
Outline
Bingo
Velocity
Tangents
![Page 4: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/4.jpg)
Hatsumon
ProblemMy speedometer is broken, but I have an odometer and a clock.How can I determine my speed?
|−4
|−3
|−2
|−1
|0
|1
|2
|3
|4
![Page 5: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/5.jpg)
Outline
Bingo
Velocity
Tangents
![Page 6: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/6.jpg)
A famous solvable problem
ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.
But what do we mean by tangent?In geometry, a line is tangent to a circle if it intersects the circle inonly one place.
•
![Page 7: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/7.jpg)
A famous solvable problem
ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.
But what do we mean by tangent?
In geometry, a line is tangent to a circle if it intersects the circle inonly one place.
•
![Page 8: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/8.jpg)
A famous solvable problem
ProblemGiven a curve and a point on the curve, find the line tangent tothe curve at that point.
But what do we mean by tangent?In geometry, a line is tangent to a circle if it intersects the circle inonly one place.
•
![Page 9: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/9.jpg)
Towards a definition of tangent
This doesn’t work so well for general curves, though:
Is this a tangent line?
Is this a tangent line?•
We need to think of tangency as a “local” phenomenon.
![Page 10: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/10.jpg)
Towards a definition of tangent
This doesn’t work so well for general curves, though:
Is this a tangent line?
Is this a tangent line?•
We need to think of tangency as a “local” phenomenon.
![Page 11: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/11.jpg)
Towards a definition of tangent
This doesn’t work so well for general curves, though:
Is this a tangent line?
Is this a tangent line?•
We need to think of tangency as a “local” phenomenon.
![Page 12: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/12.jpg)
Towards a definition of tangent
This doesn’t work so well for general curves, though:
Is this a tangent line?
Is this a tangent line?•
We need to think of tangency as a “local” phenomenon.
![Page 13: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/13.jpg)
Tangent
A line L is tangent to a curve C at a point P if
I L and C both go through P, and
I L and C have the same “slope” at P.
Slope of L = “m” in y = mx + b
=rise
run
Slope of C at a ≈ f (x)− f (a)
x − awhere x ≈ a
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Tangent as a limiting process
I To find the tangent line through a curve at a point, we drawsecant lines through the curve at that point and find the linethey approach as the second point of the secant nears the first.
I For instance, it appears the tangent line to y =√
x through(4, 2) has slope 0.25.
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Tangent as a limiting process
I To find the tangent line through a curve at a point, we drawsecant lines through the curve at that point and find the linethey approach as the second point of the secant nears the first.
I For instance, it appears the tangent line to y =√
x through(4, 2) has slope 0.25.
![Page 16: Lesson 1: The Tangent and Velocity Problems](https://reader036.vdocuments.net/reader036/viewer/2022082511/547c85acb47959c5508b46c9/html5/thumbnails/16.jpg)
Same thing!
The infinitesimal rate of change calculation is the same in bothcases: finding velocities or finding slopes of tangent lines.
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General rates of change
I The rate of change of f (t) at time t1 = the slope of y = f (t)at the point (t1, f (t1)).
I The units areunits of f (t)
units of t.