Slope Calculation
[Play
Video] 4½ minutes: Approximating
Slope of a tangent line, or taking the approximation to
Limit, when possible, to give a definition of the slope
of a tangent. Saying how to compute or approximate a
number or quantity defines.
So far the slope to a curve y = f(x) at a point
(x1,y1) = (x1,f(x1)) has been physically or graphically
associated with the slope of a short ski whose
midpoint touches a smooth (not too bumpy) curve at the point (x1,f(x1)). The
following diagram shows or suggests how the slope of such a ski resting
on the curve at the point (x1,y1) could be
approximated by the slope of a short chord joining
(x1,y1) to a nearby second point
(x2,y2) = (x2,f(x2)) on the curve.
 h
The function f(x) is assumed to be continuous at x1
- without jumps or other discontinuity there.
Consider the following.
The chord or line segment joining the point (x1,y1) to the point
(x2,y2) = (x2,f(x2) has slope
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mchord = |
y2-y1
x2-x1
|
= |
Dy
Dx
|
|
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and equation y = mchord(x-x1)+ y1. When the ski travels between x = x1 and x = x2, its
slope
is (or should be) approximated by the slope mchord of the
chord, alias line segment.
We suppose the point (x1,y1) is fixed in place.
In other words, suppose it is not moving. We further suppose
the point (x2,y2) = (x2,f(x2)) moves along the curve y = f(x)
towards the point (x1,y1). The
slope m of the line segment through these two points
should approach the slope mski = f¢(x1)
of a ski on the curve at (x1,y1).
In the motion just described, as the point
(x2,y2) = (x2,f(x2)) moves along the curve y = f(x)
towards the point (x1,y1), the abscissa x2 should
move closer to x1. The difference Dx = x2-x1
should also become closer and closer to zero. Thus we
expect the approximation
mski »
|
Dy
Dx
|
= |
y2-y1
x2-x1
|
= |
f(x2)-f(x1)
x2-x1
|
|
|
to improve when (x2,y2) = (x2,f(x2)) approaches
(x1,y1) = (x1,f(x1)) and/or as x2 approaches x1.
The continuity of f(x) at x1 implies
the moving point (x2,y2) = (x2,f(x2)) will
approach the non-moving, that is fixed point, (x1,y1) = (x1,f(x1)) when the abscissa x2
approaches x1 or equivalently, when Dx = x2-x1 approaches 0.
Note the arrow ® will be employed as shorthand
for the phrase approaches or goes to.
If the graphical and physical expectations hold, then
mski = f¢(x1) should be the limiting value of [(Dy )/(Dx)]
as Dx ® 0.
The better and better calculation of this limit should
provide an arithmetic means for approximating the expected slope of
the ski with greater and greater accuracy to an arbitrary
number of decimal places. The
limiting value of the segment slope should equal that of
the ski. This provides the computational definition
and the mathematical one as well. See the next section.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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[ Back ] [ Home ] [ Next ]
Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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