# How do I draw tangent lines?

Jun 11, 2017

See explanation...

#### Explanation:

Given a function $f \left(x\right)$ which is smooth enough in the neighbourhood of $x = a$, the tangent is a line through $\left(a , f \left(a\right)\right)$ which touches the graph of $f \left(x\right)$ at $\left(a , f \left(a\right)\right)$, but does not cross the graph within a short distance of that point.

The tangent line has the same slope as the function at that point.

What do we mean by slope of a function at a point?

If it exists, then it is the limit of the slope of lines through $\left(a , f \left(a\right)\right)$ and $\left(a + \delta , f \left(a + \delta\right)\right)$ as $\delta \to 0$.

So given the graph of $f \left(x\right)$ in the neighbourhood of $\left(a , f \left(a\right)\right)$, you can place a ruler against $\left(a , f \left(a\right)\right)$ and rotate it about that point until it no longer cuts the curve of the function. Then the ruler indicates the tangent:

graph{(y-x^3+2x)(x-y-2.001) = 0 [-0.591, 1.909, -1.52, -0.27]}