A tangent line is a local model for a differentiable function.
For f(x)=√x and a>0, f′(a)=1/(2√a). The tangent approximation is exact at x=a and has the same first derivative there. Because f″ is negative for x>0, the curve lies below this tangent. Accuracy improves as x approaches a.
A local approximation is not a global replacement for a function.