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.

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):

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

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

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.