First derivatives and tangents.

Let $ \mathcal{C}$ be the graph of a function $ f$ defined on a neighborhood of $ x_0$ . we denote by $ A$ the point whose coordinates are $ (x_0, f(x_0)$ . Through $ A$ draw a line; suppose that this line intersects $ \mathcal{C}$ at another point $ M$ . When $ M$ gets arbitrarily close to $ A$ , then the line $ (AM)$ sometimes gets ``closer to a limit position'', which will be called the tangent to $ \mathcal{C}$ at $ A$ .

Figure 2: The limit position of a secant to a curve at a given point.
\begin{figure}\mbox{\epsfig{file=cubic_tangent.eps,height=4cm}}\end{figure}

Actually , this is exactly what is yielded by the linear approxianmation of a function $ f$ at a point $ x_0$ (v.s. Definition 2.1):

$\displaystyle f(x_0+h)=f(x_0)+Ah+h\; \varepsilon(h) $ (6.1)

Denote $ x=x_0+h$ , then the affine part of the right-hand expression in Equation (3) is $ f(x_0)+(x-x_0)f'(x_0)$ . The equation $ y=f(x_0)+(x-x_0)f'(x_0)$ is the equation of a line; this is te tangent to the graph of $ f$ at the point whose $ x-$ coorsinate is equal to $ x_0$ .

Theorem 6.3.1   If $ f$ is differentiable at $ x_0$ , then $ \mathcal{C}$ has a tangent at $ A(x_0, f(x_0)$ ; the slope of this tangent is equal to $ f'(x_0)$ .

The equation of the tangent to $ \mathcal{C}$ at $ A(x_0, f(x_)0$ is

$\displaystyle y-f(x_0)=f'(x_0)(x-x_0)$    

Example 6.3.2   Let $ f(x)=x^3-3x^2+x+1$ . Then $ f'(x)=3x^2-6x+1$ . If $ x_0=0$ , we have $ f'(0)=1$ and the equation of the tangent at the point $ A(0,1)$ is $ y=x+1$ .

Figure 3: Tangent to a curve at a point.
\begin{figure}\mbox{\epsfig{file=cubic_tangent2.eps,height=4cm}}\end{figure}

Noah Dana-Picard 2007-12-28