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.
Best Answer
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.