Slope of a Curve at a Point
1
$begingroup$
How can a point on a curve have a “slope”? I do not understand this. I thought slopes were only features of straight lines.
calculus derivatives
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
$endgroup$
add a comment |
1
$begingroup$
How can a point on a curve have a “slope”? I do not understand this. I thought slopes were only features of straight lines.
calculus derivatives
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
$endgroup$
1
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
1
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49
add a comment |
1
1
1
1
$begingroup$
How can a point on a curve have a “slope”? I do not understand this. I thought slopes were only features of straight lines.
calculus derivatives
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
$endgroup$
How can a point on a curve have a “slope”? I do not understand this. I thought slopes were only features of straight lines.
calculus derivatives
calculus derivatives
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
asked Nov 26 '18 at 3:40
InterestingggInterestinggg
61
61
1
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
1
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49
add a comment |
1
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
1
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49
1
1
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
1
1
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49
add a comment |
1 Answer
1
active
oldest
votes
0
$begingroup$
The slope is not defined for a point. "The slope of a point" you hear is the slope of the tangent at that point.
Any curve you might encounter has numerous points on it, there is a tangent for every one of that point. So the slope of the tangent on the curve at that particular point can be said as "slope of the point" informally.
Mathematically, it is just defined as the slope of the tangent on that point.
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
$endgroup$
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
add a comment |
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0
$begingroup$
The slope is not defined for a point. "The slope of a point" you hear is the slope of the tangent at that point.
Any curve you might encounter has numerous points on it, there is a tangent for every one of that point. So the slope of the tangent on the curve at that particular point can be said as "slope of the point" informally.
Mathematically, it is just defined as the slope of the tangent on that point.
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
$endgroup$
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
add a comment |
0
$begingroup$
The slope is not defined for a point. "The slope of a point" you hear is the slope of the tangent at that point.
Any curve you might encounter has numerous points on it, there is a tangent for every one of that point. So the slope of the tangent on the curve at that particular point can be said as "slope of the point" informally.
Mathematically, it is just defined as the slope of the tangent on that point.
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
$endgroup$
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
add a comment |
0
0
0
$begingroup$
The slope is not defined for a point. "The slope of a point" you hear is the slope of the tangent at that point.
Any curve you might encounter has numerous points on it, there is a tangent for every one of that point. So the slope of the tangent on the curve at that particular point can be said as "slope of the point" informally.
Mathematically, it is just defined as the slope of the tangent on that point.
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
$endgroup$
The slope is not defined for a point. "The slope of a point" you hear is the slope of the tangent at that point.
Any curve you might encounter has numerous points on it, there is a tangent for every one of that point. So the slope of the tangent on the curve at that particular point can be said as "slope of the point" informally.
Mathematically, it is just defined as the slope of the tangent on that point.
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
answered Nov 26 '18 at 3:42
PradyumanDixitPradyumanDixit
832214
832214
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
add a comment |
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
2
2
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
$begingroup$
So in other words “the slope of a point of a curve” is just short for “the slope of the tangent to that point” am I correct in saying that?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:47
1
1
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
$begingroup$
But something else is confusing to me. If slope is not defined for a single point, then how can a point have an instantaneous rate of change? Is it because the instaneous rate is the same as the slope of the tangent line?
$endgroup$
– Interestinggg
Nov 26 '18 at 3:48
1
1
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
Right. So we associate each point with a slope which we compute in the context of the curve. The point itself is just a point. It only has a meaningful slope in the context of the curve. The point doesn't have an instantaneous rate of change. It's more like the curve has a rate of change and we can even compute an instantaneous rate of change which we can associate with each point on the curve.
$endgroup$
– Mason
Nov 26 '18 at 3:55
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
@Interestinggg sorry I was in my class, couldn't see the comment. The point does not have the rate of change, the graph of the curve represents the playground and tangents on those points actually have changing slopes, so rate of change of y with respect to x is given by slope of the tangent of the curve at the point $(x,y)$
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
$begingroup$
This is exactly what @Mason said, but just in other words, for better understanding.
$endgroup$
– PradyumanDixit
Nov 26 '18 at 13:08
add a comment |
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1
$begingroup$
desmos.com/calculator/kkdiaarmta. @interestinggg. You are correct in many ways. A slope is a well-defined concept on a line. So we can approach the slope along an arbitrary curve by creating a secant line and then we define the slope at a given point via a tangent line.
$endgroup$
– Mason
Nov 26 '18 at 3:44
$begingroup$
@Mason nice way to show :)
$endgroup$
– PradyumanDixit
Nov 26 '18 at 3:45
1
$begingroup$
@PradyumanDixit. Thanks! I am proud of this one. Did you toggle all of the folders on and off??!?! desmos.com/calculator/nge0kpus3a. And you can change the function! Ok. I am a little too proud of that one...
$endgroup$
– Mason
Nov 26 '18 at 3:49