plotting - Finding the tangent to a piecewise function

I'm new to Mathematica and I am trying to calculate the expression of the tangent to the curve

f[X_] = 
  Piecewise[{{-X^2 + 4 X - 3, X < 3}, {3 - X (Abs[X - 4]), X >= 3}}]

at x = 2.

I have to draw the function and the tangent line. I would really appreciate some help.

Answer

Well, easy. Just calulate the slope at $x=2$ and then, plot the tangent with that slope and through the point $(2, f(2))$.

The slope is calulated as the first derivative of $f(x)$ in $x=2$:

D[f[v], v] /. v -> 2

and the tangent line passing through $(2, f(2))$ with $f(2)=1$ is:

X*(D[f[v], v] /. v -> 2) + f[2]

Plotting both functions:

Plot[{Piecewise[{{-3 + 4 X - X^2, X < 3}, {3 - X Abs[-4 + X], X >= 3}}, 0],
X*(D[f[v], v] /. v -> 2) + f[2]}, {X, 0, 5}, 
PlotRange -> All, Epilog -> {Red, PointSize[0.02], Point[{2, f[2]}]}]

Comments

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plotting - Plot 4D data with color as 4th dimension

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plotting - Mathematica: 3D plot based on combined 2D graphs

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