AP Calculus AB/BC - Contextual & Analytical Applications of Differentiation

$f'(x) > 0$ → Function is increasing.

$f'(x) < 0$ → Function is decreasing.

$f'(x) = 0$ or DNE → Critical Values at $x$.

Relative Maximum: $f'(x) = 0$ or DNE, and sign of $f'(x)$ changes from $+$ to $-$.

Relative Minimum: $f'(x) = 0$ or DNE, and sign of $f'(x)$ changes from $-$ to $+$.

Absolute Max or Min: MUST CHECK ENDPOINTS ALSO

The maximum value is a $y$-value.

$f''(x) > 0$ → Function is concave up.

$f''(x) < 0$ → Function is concave down.

$f'(x) = 0$ and sign of $f''(x)$ changes → Point of inflection at $x$.

Relative Maximum: $f''(x) < 0$

Relative Minimum: $f''(x) > 0$

To write the equation of a tangent line at a point, you need:

• A slope (derivative).
• A point $(x_1, y_1)$.

$$y - y_1 = m (x - x_1)$$