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







share|cite|improve this question









$endgroup$








  • 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


















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







share|cite|improve this question









$endgroup$








  • 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
















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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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
















  • 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












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.






share|cite|improve this answer









$endgroup$









  • 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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3013798%2fslope-of-a-curve-at-a-point%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






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.






share|cite|improve this answer









$endgroup$









  • 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
















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.






share|cite|improve this answer









$endgroup$









  • 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














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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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














  • 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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3013798%2fslope-of-a-curve-at-a-point%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

How to change which sound is reproduced for terminal bell?

Image
3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
Read more

Can I use Tabulator js library in my java Spring + Thymeleaf project?

Image
0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
Read more

Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

Image
up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx...
Read more