Question

What is a tangent line?

Answer 1

The tangent line at a point is the straight line that juts touches the curve at that point.
Thats the definitio by wikipedia, but I believe that's a pretty good definition. The main ideia is that the tangent line touches a curve in one single point.
Hope youo understand the ideia. Bye. =)

Answer 2

To find the tangent line \(\large\color{blue}{\displaystyle { l(x)}}\) to any curve \(\large\color{blue}{\displaystyle { f(x)}}\) at \(\large\color{slate}{\displaystyle { x=\color{blue}{a}}}\) , you perform the following steps.

STEP 1

You find the y-coordinate of the point, by plugging in the \(\large\color{slate}{\displaystyle { \color{blue}{a}}}\) into the \(\large\color{blue}{\displaystyle { f(x)}}\).
\(\large\color{slate}{\displaystyle {(}}\)So your y-coordinate will simply be \(\large\color{blue}{\displaystyle { f(a)}}\) (for whatever it will be when you calculate it.\(\large\color{slate}{\displaystyle {)}}\)


STEP 2
Differentiate (take the derivative of) the curve \(\large\color{blue}{\displaystyle { f(x)}}\).
(Remember, the derivative of the function is it's slope, and slope is exactly what we are looking for over here.) \(\large\color{slate}{\displaystyle {(}}\) the derivative will be \(\large\color{blue}{\displaystyle { f'(x)}}\) \(\large\color{slate} {\displaystyle)}\)


STEP 3
Once you found the slope (which is the derivative of the function) you want to find the slope of the function at a specific point, and in your case it is a slope of the function when \(\large\color{slate}{\displaystyle { x=\color{blue}{a}}}\),
so plug in \(\large\color{slate}{\displaystyle { \color{blue}{a}}}\) into the \(\large\color{blue}{\displaystyle { f'(x)}}\). \(\large\color{slate} {\displaystyle(}\)Which will simply be \(\large\color{blue}{\displaystyle { f'(a)}}\). \(\large\color{slate} {\displaystyle)}\)


STEP 4
1. once you found \(\large\color{blue}{\displaystyle { f'(a)}}\) (the slope of the tangent)
2. and once you know the point \(\large\color{blue}{\displaystyle { \left( a,\color{white}{\LARGE |}f(a)\right)}}\)
3. plug in your information into a point slope form of the line.

Answer 3

\(\large\color{slate}{\displaystyle\bf y-y_{_1}=m\left( x-x_{_1}\right)}\)

plugging in...

\(\large\color{slate}{\displaystyle\bf y~-~f(a)~=~f'(a)~\left( x~-~a\right)}\)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Or, alternative forms,

\(\large\color{slate}{\displaystyle\bf y~=~xf'(a)~-~af'(a)~+~f(a)}\)
\(\large\color{slate}{\displaystyle\bf y~=~xf'(a)~+~f(a)~-~af'(a)}\)

where , comparing to \(\large\color{slate}{\displaystyle\bf y~=~mx~+b}\)
\(\large\color{slate}{\displaystyle\bf m=f'(a)}\)
\(\large\color{slate}{\displaystyle\bf b=f(a)-af'(a)}\)

Answer 4

Note that tangent line has same slope as the curve it touches. It has only one common point at the touching point and it's near region but can have other common points if the curve is "kinky".

An example is curve y = x^3. If you take point (-1,-1) and draw a tangent there you find that the line crosses the curve at (2,8). So the tanget line has two common points with the curve but it's still the tanget as it's common slope point has no other common points in it near region.

Tangent line can also be a tangent to multiple points of the curve if the curve has other common points with the line and has same slope there. Simple example is the line y=1 for curve y=cos(x) where this happens, but there are others more complicated situations.

So, in summary: Tangent line is only defined by it's slope at the touching point and it's near region, outside that near region anything goes.

Answer 5

Tangent line of a curve is the straight line drawn at any point on the curve which shows the direction of the curve at that point.