Questions regarding tangent line

Hi, I have 3 tiny questions

How you calculated the slop of the tangent line, also how did you decide the slop if negative or positive ? is there any formula ?

If you see my attached photo, the line on my curve, why can’t that line be a tangent line ? since its also not touching to the curve at any other point

  1. in this below attached photo, what could be the tangent line on the selected red point (asking because if we draw a line (maybe having a positive slope), It will touch at the top of that curve

Thanks

For your points:

  1. You can differentiate the equation of that curve with respect to its parameter to find the slope.

  2. That line in your image, however, is not a tangent line. A tangent line always touches a curve at a single point.

  3. Yes, it will have a positive slope, which may look like a line that rises as it moves to the right:
    image

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A tangent touches at one point and does not cross the line.

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Thanks for your reply

I have still questions regarding 2nd question-answer

That line in your image, however, is not a tangent line. A tangent line always touches a curve at a single point.

Yes, if you see it touches at only one point (at the bottom of the curve)

regarding 3rd question-answer

Yes, it will have a positive slope, which may look like a line that rises as it moves to the right:

I have exnteded that line, see it touches to the bottom of the curve, here its touching at the bottom of that curve, 2nd time, how it can be a tangent line then ?

Thanks

I didn’t mention one more thing, which is that it should not cross the curve, as Tom said.

It still remains a tangent line for the ‘red point’ because it does not cross the curve at that point.


PS: There is a gap between those two lines, but I am assuming it to be a single tangent line.

A tangent is local to a specific region of a curve.

It is not global.

A definition like Tangent is a line that only touches the curve at a single point, is not quite right. It can be seen as the property of the tangent but not the definition.
I don’t have a formal definition too but a good one.

Let’s start from here.
What is a slope of a line y=f(x)? It is change in y with respect to change in x.
Screenshot from 2023-07-04 06-33-45

For a line, the rate of change is constant. But in the case of a curve y = f(x), the rate of change in y with respect to x is not constant (that’s why it is a curve, else it would have been a straight line)

So, how do we find the rate of Change for a curve?
Well, we would need Tangent. Just take two points on the curve and join them with a straight line. And you can calculate the average rate of range between those two points same as for the above line.

Screenshot from 2023-07-04 06-35-16

But what if we want to know the instantaneous rate of change at a particular point only?
We use limits for this purpose. Think of it like that.
Bring the two points from the above diagram closer and closer and closer until they are the kind of same but with an infinitesimally small distance between them.
So now, when you connect these two points and draw a straight line. That is your tangent.

In your first diagram, you are not taking the two points on the curve. You can’t actually draw a unique line with a single point. You need at least two. And in your case, the second point is not on the curve but outside. Your line is not showing the rate of change in y with respect to change in x. So just because it is touching the curve at one point doesn’t make it tangent.

If seen from the limits perspective, the tangent is not actually touching at one point but two. But the distance between them tends to be zero which means the distance is getting closer and closer to zero. Something like 0.00000000…001.

Joining the lines gives us the tangent. So then It doesn’t matter if it touches the complete curve at one point or two or ten.

Given above a graph for same. Try changing the value of “h” on that page and see yourself how you get a tangent.

I think I made it clear. if not please point out the line. I’d try to explain even better.
But always try to understand what anything is than to follow the rules directly.

Great explanation, @pandeyganesha! Thank you!

Just to give ourselves some intuitive view of tangents, the following GIF was “stolen” from Wikipedia. Here is the source.

Graph_of_sliding_derivative_line

The tangent line that represents the slope at a point touches that point and is also “parallel to that point”. By being “parallel to that point”, I am trying to say what @pandeyganesha has explained - that we find two points very very very very near to the point of interest and form a line with those two points, then the slope is parallel to such line.

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