Is Slope Purely a Geometric Concept and is it Defined for a Single Point?
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Is “slope” purely a geometric concept? For instance can we talk about the “slope” of a function or only the “slope” of a straight line? Furthermore, can we define “slope” for a single point? We often say the slope of the curve at a point, but how can a single point have a slope?
calculus derivatives
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
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add a comment |
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Is “slope” purely a geometric concept? For instance can we talk about the “slope” of a function or only the “slope” of a straight line? Furthermore, can we define “slope” for a single point? We often say the slope of the curve at a point, but how can a single point have a slope?
calculus derivatives
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
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3
$begingroup$
The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
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– MJD
Nov 26 '18 at 22:13
add a comment |
$begingroup$
Is “slope” purely a geometric concept? For instance can we talk about the “slope” of a function or only the “slope” of a straight line? Furthermore, can we define “slope” for a single point? We often say the slope of the curve at a point, but how can a single point have a slope?
calculus derivatives
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
$endgroup$
Is “slope” purely a geometric concept? For instance can we talk about the “slope” of a function or only the “slope” of a straight line? Furthermore, can we define “slope” for a single point? We often say the slope of the curve at a point, but how can a single point have a slope?
calculus derivatives
calculus derivatives
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
asked Nov 26 '18 at 21:41
J. SmithJ. Smith
241
241
3
$begingroup$
The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
$endgroup$
– MJD
Nov 26 '18 at 22:13
add a comment |
3
$begingroup$
The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
$endgroup$
– MJD
Nov 26 '18 at 22:13
3
3
$begingroup$
The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
$endgroup$
– MJD
Nov 26 '18 at 22:13
$begingroup$
The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
$endgroup$
– MJD
Nov 26 '18 at 22:13
add a comment |
2 Answers
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oldest
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The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
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add a comment |
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[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]
We can define the slope
of a curve
at a point
as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.
Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:
- a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
- the slope at a point is the slope of one of these line segments.
Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
An infinitely short one wouldn't have two distinct endpoints.
No luck if we go back to Euclid, either: his definition was:
A line is breadthless length.
An infinitely short one wouldn't have length.
In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".
We simply define the slope of the curve at a point as being that left-over slope.
If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
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active
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votes
$begingroup$
The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
$endgroup$
add a comment |
$begingroup$
The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
$endgroup$
add a comment |
$begingroup$
The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
$endgroup$
The slope is for the curve (an infinite collection of points). It just happens to vary from point to point. We can use the slope as a synonym for derivative.
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
answered Nov 26 '18 at 21:45
AndreiAndrei
12.6k21128
12.6k21128
add a comment |
add a comment |
$begingroup$
[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]
We can define the slope
of a curve
at a point
as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.
Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:
- a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
- the slope at a point is the slope of one of these line segments.
Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
An infinitely short one wouldn't have two distinct endpoints.
No luck if we go back to Euclid, either: his definition was:
A line is breadthless length.
An infinitely short one wouldn't have length.
In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".
We simply define the slope of the curve at a point as being that left-over slope.
If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
$endgroup$
add a comment |
$begingroup$
[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]
We can define the slope
of a curve
at a point
as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.
Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:
- a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
- the slope at a point is the slope of one of these line segments.
Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
An infinitely short one wouldn't have two distinct endpoints.
No luck if we go back to Euclid, either: his definition was:
A line is breadthless length.
An infinitely short one wouldn't have length.
In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".
We simply define the slope of the curve at a point as being that left-over slope.
If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
$endgroup$
add a comment |
$begingroup$
[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]
We can define the slope
of a curve
at a point
as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.
Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:
- a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
- the slope at a point is the slope of one of these line segments.
Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
An infinitely short one wouldn't have two distinct endpoints.
No luck if we go back to Euclid, either: his definition was:
A line is breadthless length.
An infinitely short one wouldn't have length.
In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".
We simply define the slope of the curve at a point as being that left-over slope.
If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
$endgroup$
[Note: I'm going to use line and line segment interchangeably here, to save typing. I mean it to have a non-infinite length.]
We can define the slope
of a curve
at a point
as the slope of the unique tangent to the curve at that point. If the tangent isn't unique, then the curve doesn't have a defined slope there.
Where it gets conceptually difficult is when we try to say the following, which I imagine is what bothers you:
- a curve is made up of infinitely short line segments, which look the same as points since they have zero length.
- the slope at a point is the slope of one of these line segments.
Well, by accepted definitions, there's no such thing as an infinitely short line segment. For example the Wikipedia one:
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.
An infinitely short one wouldn't have two distinct endpoints.
No luck if we go back to Euclid, either: his definition was:
A line is breadthless length.
An infinitely short one wouldn't have length.
In geometric representations of differentiation, calculus sidesteps the problem: instead of having an infinitely short line, it treats the slope at a point as the limit of the slope of a line whose length approaches $0$. You can imagine it as "the slope the infinitely short line would have if it could exist", or maybe "the slope left over after the line has disappeared".
We simply define the slope of the curve at a point as being that left-over slope.
If the curve happens to be the graph of a function drawn using $x$–$y$ coordinates, then its slope at a point represents the function's derivative there. Taken literally, the "slope of the function" at a point would be a geometric metaphor based on treating the curve as the actual function. But it's become a standard alternative name for the derivative itself.
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
edited Jan 2 at 1:30
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
answered Jan 2 at 1:19
timtfjtimtfj
2,458420
2,458420
add a comment |
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The point doesn't have a slope, and nobody says it does. The curve has a slope at a point. Consider: Noon on Wednesday is a particular time. We don't normally think of it as having a sound. But Puddles the puppy might have a sound at noon on Wednesday.
$endgroup$
– MJD
Nov 26 '18 at 22:13