Why does it seem like that a tangent line of an odd-degree polynomial function crosses the curve at more than one point?

Question

This question has bothered me for a long time. I know that a tangent line only crosses a curve at one specific point. However, consider this:

Let $f(x)=x^3$
The derivative of this function is $f^\prime(x)=3x^2$
Suppose that I want to know the equation of the tangent line at $x=1$. To determine the slope, I just need to plug $x=1$ into $f^\prime(x)$ :
$f^\prime(1)=3(1)^2=3$

Therefore, the slope at $x=1$ of $f(x)$ is $3$.
Using the slope formula, I get $(y-1)=3(x-1)$
Solving for $y$, I get $y=3x-2$

Therefore, the tangent line equation of the cubic curve at $x=1$ is $y=3x-2$

I would think that the tangent line will only intersects the cubic curve at a single point, but when I typed this into the Desmos Calculator , it intersects on the curve at two points $(1,1)$ and $(-2,-8)$.

This is actually different for a quadratic function. Consider this:

Let $f(x)=x^2$
Then $f^\prime(x)=2x$

Suppose that I want to find the slope of the function at $x=1$:
$f^\prime(1)=2(1)=2$
The equation of the tangent line at the point when $x=1$ is
$(y-1)=2(x-1)$
$y=2x-1$

When I typed this into the Desmos Calculator , it shows me that the only solution to this pair of system of equation is $(1,1)$

Why did this happen? Am I misunderstanding something? Thank you for your time.

Answer 1

I think your misunderstanding is that "a tangent line only crosses a curve at one specific point". If a line $L$ is tangent to a curve $C$ at a point $P$, then $L$ will meet $C$ at $P$ and will have the same direction. That's all. There is nothing in general to say whether or not $L$ will also meet $C$ at some other point, and as your example shows, this may well happen with a cubic.

It won't happen with a quadratic since a quadratic always "curves in the same direction", therefore moving away from the tangent. But it can also happen with a sine curve, for example - draw some pictures and you will easily find a tangent which intersects the curve again at one other point, or two, or in fact many - and it can happen with various other curves too.

In fact, a line will always meet a cubic curve in exactly three points (if you allow complex solutions: at most three, if not). In saying this, you have to count a tangent as two meeting points, and the third will usually be somewhere else. In the case where $C$ is $y=x^3$ and $P$ is the origin, the tangent $L$ is $y=0$. Here $P$ is also a point of inflection and all three meeting points are the same.

Hope this helps.

Answer 2

VISUALIZATION :

Here is a Degree 4 Polynomial :

There are atleast 3 types of tangents here.

Type 1 : We can see that the lower thin Purple line is tangent & touches only once.

That is not the only type of tangent which might occur.

Type 2 : The thick Purple Solid line is a tangent at around $(1.9,-25)$ & amazingly , that tangent line touches the curve again tangentially at around $(-1.9,-2)$ !
Hence tangents can touch the curve multiple times.

Type 3 : The thick Purple Dotted line is a tangent at around $(-0.1,-1)$ & it intersects the curve at around $(-2.9,22)$ & around $(-2.4,-20)$

Over-all : A tangent may touch exactly once or a tangent might be tangential multiple times or a tangent might intersect the curve multiples times.

Here is a Degree 6 curve , where we might want to identify such types of tangents :

It has a tangent line which is tangential at 2 Points & that tangent line intersects the curve at 2 other Points !

[[ IMAGES GENERATED COURTESY OF WOLFRAM ONLINE TOOL AND THEN MODIFIED ]]

ADDENDUM :

(A) Why is it claimed that a tangent touches only once ?
(1) It is a wrong claim , based on faulty thinking.
(2) It occurs because the Derivation of the tangent line uses a technique which might misleading into that thinking.
Basically , that technique uses a line drawn to intersect a curve at 2 Points. then the two Points are moved closer (by changing the Parameters) & then the tangent line is claimed to be the limit when the 2 Points coincide to become a Single Point. That is true , though it makes some faulty thinking that the tangent will not touch or intersect later. It only makes the local intersection a tangent when the 2 Points coincide & merge to a Single Point.

(B) Degree of Polynomial is almost immaterial , such tangents might occur for Degree 3 , 4 , 5 , 6 ....

(C) Though it is not universal , there are certain classes of curves where tangent line will touch exactly once : Eg , Circle , Ellipse , Parabola

(D) In 3D Case , we have the concept of tangent plane , where tangency & intersection can occur like the 2D Case tangent line.