Finding the speed of a moving object (without a speedometer) and finding the slope of a line tangent to a curve are two interesting problems. It turns out there are models of the same process.

1. Section 2.1
The Tangent and Velocity Problems
Math 1a
February 1, 2008
Announcements
Grab a bingo card and start playing!
Syllabus available on course website
Homework for Monday 2/4:
Practice 2.1: 1, 3, 5, 7, 9
Turn-in 2.1: 2, 4, 6, 8
Complete the ALEKS initial assessment (course code
QAQRC-EQJA6)

2. Outline
Bingo
Velocity
Tangents

3. Outline
Bingo
Velocity
Tangents

4. Hatsumon
Problem
My 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

5. Outline
Bingo
Velocity
Tangents

6. A famous solvable problem
Problem
Given a curve and a point on the curve, find the line tangent to
the curve at that point.

7. A famous solvable problem
Problem
Given a curve and a point on the curve, find the line tangent to
the curve at that point.
But what do we mean by tangent?

8. A famous solvable problem
Problem
Given a curve and a point on the curve, find the line tangent to
the 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 in
only one place.
•

9. Towards a definition of tangent
This doesn’t work so well for general curves, though:

10. Towards a definition of tangent
This doesn’t work so well for general curves, though:
Is this a tangent line?

11. 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?

12. 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.

13. Tangent
A line L is tangent to a curve C at a point P if
L and C both go through P, and
L and C have the same “slope” at P.
Slope of L = “m” in y = mx + b
rise
=
run
f (x) − f (a)
Slope of C at a ≈ where x ≈ a
x −a

14. Tangent as a limiting process
To find the tangent line through a curve at a point, we draw
secant lines through the curve at that point and find the line
they approach as the second point of the secant nears the first.

15. Tangent as a limiting process
To find the tangent line through a curve at a point, we draw
secant lines through the curve at that point and find the line
they approach as the second point of the secant nears the first.
√
For instance, it appears the tangent line to y = x through
(4, 2) has slope 0.25.

16. Same thing!
The infinitesimal rate of change calculation is the same in both
cases: finding velocities or finding slopes of tangent lines.

17. General rates of change
The rate of change of f (t) at time t1 = the slope of y = f (t)
at the point (t1 , f (t1 )).
units of f (t)
The units are .
units of t