A tangent to a curve

From the secant to the tangent

As h approaches 0, the secant line passing through (a, f(a)) and (a+h, f(a+h)) approaches a limit position: the tangent to the curve of f at the point with abscissa a.

Equation of the tangent

If f is differentiable at a, the equation of the tangent at the point A(a, f(a)) is:

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

This straight line passes through point A and has a slope of f'(a).

Practical example

Let’s take f(x) = x² again, with a = 3. We have f(3) = 9 and we find that f'(3) = 6. The equation of the tangent line at the point (3, 9) is therefore:

y = 6 * (x - 3) + 9 = 6x - 18 + 9 = 6x - 9

If x = 4, the tangent gives y = 24 – 9 = 15, whereas f(4) = 16: the tangent does indeed follow the curve closely near x = 3, but diverges as we move further away.

Determining the sign using the tangent

• If f'(a) > 0, the tangent line rises: f is locally increasing at a.
• If f'(a) < 0, the tangent line falls: f is locally decreasing at a.
• If f'(a) = 0, the tangent line is horizontal: a is a point where f may have an extremum (local maximum or minimum).

A common pitfall

A horizontal tangent line (f'(a) = 0) does not always guarantee an extremum: consider f(x) = x³ at a = 0, where f'(0) = 0, but the curve crosses its tangent line without reaching a maximum or minimum (this is a point of inflection). You must always check the change in sign of f' before drawing any conclusions.