About GitHub Coach Log in Start reading

Mathematics · Glossary

What is Tangent?

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

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

y=f(a)+f′(a)(x−a).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 AA and B=(a+h,f(a+h))B = \bigl(a+h, f(a+h)\bigr) has slope f(a+h)−f(a)h\frac{f(a+h)-f(a)}{h}. As BB slides towards AA, the secant tilts into the tangent, whose slope is f′(a)f'(a).

Examples

Example 12.7

For f(x)=x2f(x) = x^2 at a=1a = 1: f(1)=1f(1) = 1 and f′(1)=2f'(1) = 2 (Example 12.4), so the tangent is

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

This is the line d2d_2 found by pure algebra in Exercise 10.11.

Read in context →

Related terms