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Mathematics · 术语表

什么是 Tangent？

Definition 12.6 High School Mathematics · 第 12 章 — Differentiation: A First Course

If $f$ is differentiable at $a$ , the tangent to the curve at the point $A = \bigl(a, f(a)\bigr)$ is the line through $A$ with slope $f'(a)$ . Its equation is

y = f(a) + f'(a)(x - a).

The secant through A and B = (a+h, f(a+h) ) has slope f(a+h)-f(a)/h. As B slides towards A, the secant tilts into the tangent, whose slope is f'(a). — The secant through $A$ and $B = \bigl(a+h, f(a+h)\bigr)$ has slope $\frac{f(a+h)-f(a)}{h}$ . As $B$ slides towards $A$ , the secant tilts into the tangent, whose slope is $f'(a)$ .

示例

Example 12.7

For $f(x) = x^2$ at $a = 1$ : $f(1) = 1$ and $f'(1) = 2$ (Example 12.4), so the tangent is

y = 1 + 2(x - 1) = 2x - 1 .

This is the line $d_2$ found by pure algebra in Exercise 10.11.

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