What Does "Tangent to a Point on a Curve" Mean?

1,247

Basically, a tangent line through a given point on a curve is the boundary of a half-plane which contains a segment of the curve to either side of the point, and for which the point itself is on the boundary.

I will give you an example:

Say the curve $y = 5 - x^2$ at the point $(1,4)$ on the curve and you want to find the slope of line tangent to a curve at that given point. You would basically utilize the derivative, $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ to find the slope of the tangent line to the curve at the point $(1, 4)$, where $$y = f(x) = 5 - x^2$$ So, $$\begin{align} f'(x) =\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} & = \lim_{h \to 0} \frac{\Big(5 - (x + h)^2\Big) - (5 - x^2)}{h} \\ \\ & = \lim_{h\to 0}\frac{5 - (x^2 + 2hx + h^2) - 5 + x^2}{h} \\ \\ & = \lim_{h\to 0}\frac {-2hx - h^2}{h} \\ \\ & = \lim_{h\to 0}\frac{-h(2x + h)}{h} \\ \\ & = \lim_{h\to 0} -(2x + h)\\ \\ f'(x) & = -2x \end{align}$$

Now, at $(1, 4), \;x = 1$, so the slope at that point is given by $f'(1) = -2(1) = -2$.

Here is a depiction of how Wolfram Alpha shows it: (Image compliments of WolframAlpha).

enter image description here

If you want to go further and find the actual equation of the line then, you can find the equation line tangent to $y = 5-x^2$ at the point $(1,4)$ knowing the slope of the line is $-2$, and the fact that $(1, 4)$ lies on that line. We know the equation of a line can be written in the form $(y - y_0) = m(x - x_0)$, where $m$ represents the slope of the line, and $(x_0, y_0)$ represents a point on the line.

In our case, we've found slope to be given by $-2$, and $(x_0, y_0) = (1, 4)$. So, $$y - 4 = -2(x - 1) \iff y - 4 = -2x + 2 \iff y = 6 - 2x$$

Share:
1,247

Related videos on Youtube

Inquisitive
Author by

Inquisitive

Updated on September 23, 2022

Comments

  • Inquisitive
    Inquisitive about 1 year

    Is has been stated in an earlier answer to another question that a "point" is an undefined concept in mathematics.

    If this is true, how can we define "tangent to a point on a curve"?

    • André Nicolas
      André Nicolas almost 9 years
      You may mean tangent to a curve at a point.
    • turkeyhundt
      turkeyhundt almost 9 years
      I think this is a case where mathematical terms and plain language(english) might be blurring together. Just because a point is "undefined" doesn't mean it doesn't exist.
    • bjd2385
      bjd2385 almost 9 years
      Points certainly exist, but I believe this is really just a mixup. They likely mean the tangent line to a curve, which indicates that it is tangent to one point that is a member of the set of points making up the curve.
    • Inquisitive
      Inquisitive almost 9 years
      Is a curve made up of a string of extremely tiny circles?
    • Inquisitive
      Inquisitive almost 9 years
      I received criticism for an answer I provided elsewhere. I began to question whether or not I understood what a simple point was. As it turns out, even heavy hitters here are confused by the concept. I was hoping for something that would clarify the concept in my mind.
    • MPW
      MPW almost 9 years
      @IsaacWannabeeNewton: Yes--those extremely tiny circles are circles of radius zero, also known as "points". :/
  • Inquisitive
    Inquisitive almost 9 years
    Yagna, is it possible to have two tangent lines at that one point with each of the tangents having a different slope?
  • Admin
    Admin almost 9 years
    @IsaacWannabeeNewton No it is not possible. However the slope varies with the slightest change on the graph itself. So, if the point is say, (2,1), the slope would be completely different. The slope is basically dependent on the derivative of the function.
  • Admin
    Admin almost 9 years
    @IsaacWannabeeNewton Well not completely: it would be -4. And this is easily verifiable by the graph. As you move along the curve, the slope changes.
  • Admin
    Admin almost 9 years
    @IsaacWannabeeNewton No Problem.
  • Pete L. Clark
    Pete L. Clark almost 9 years
    "Basically, a tangent line through a given point on a curve is the boundary of a half-plane which contains a segment of the curve to either side of the point, and for which the point itself is on the boundary." This is not a bad first intuition about tangent lines. Subject to suitable differentiability and convexity conditions, it becomes a characteristic property. But it is not suitable as a definition in general: consider even that the tangent line to $f(x) = x^3$ at $x = 0$ does not satisfy your "boundary plane" property.