Tangent line equation: y = f'(a)(x − a) + f(a)

The magic formula

The tangent line to the graph of $f$ at $x = a$ has equation:

$y = f'(a) \times (x - a) + f(a)$

• Slope: $f'(a)$.
• Passes through $(a, f(a))$.

3-step method

1. Compute $f'(x)$.
2. Evaluate $f(a)$ and $f'(a)$.
3. Substitute into the formula.

Example

Tangent to $f(x) = x^2$ at $a = 3$:

$f(3) = 9$, $f'(x) = 2x$, $f'(3) = 6$.

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

Special cases

• Horizontal tangent ($f'(a) = 0$): $y = f(a)$. Often an extremum.
• Vertical tangent: $f$ is not differentiable at $a$ (e.g., $\sqrt{x}$ at $0$).

Curve vs tangent

• If $f'' > 0$ near $a$: $f$ convex ⇒ curve above the tangent.
• If $f'' < 0$: $f$ concave ⇒ curve below.
• If $f''(a) = 0$ with sign change: inflection point (curve crosses the tangent).

Application: approximation

For $x$ near $a$: $f(x) \approx f'(a)(x - a) + f(a)$.

Basis for Taylor expansions.

Common mistakes

1. Forgetting to compute $f'(a)$ (using $f(a)$ instead).
2. Writing $y = f'(a) x + f(a)$ (missing $(x - a)$).
3. Sign error in $(x - a)$ vs $(a - x)$.
4. Using $f'(x)$ unevaluated (keeping $x$ instead of plugging in $a$).

?À toi de jouer

Tangent to f(x) = x³ at a = 2?

A.y = 12x − 16B.y = 3x² − 4C.y = 12(x − 2)D.y = 8 + 12x

Valider