For questions on the tangent line, the unique straight line that is the best linear approximation to a function at a point.
If $y=f(x)$ is differentiable at $a$, the equation of the tangent line to $f$ at $(a,f(a))$ is $$ T_a(x) = f(a) + f'(a)(x-a) $$ Common uses are in the definition of differentiation and finding tangent lines to circles in geometry.
The tangent line need not touch a function locally only once. Indeed, consider $s(x) = x^3\sin(1/x)$ if $x\neq 0$, $s(0)=0$. Then $s$ is differentiable at $x=0$ with tangent line $y=0$, but this intersects $s$ infinitely often in any neighborhood of zero.
