Applications Of Derivatives – Tangents and Normal

Slope of tangent = \( \frac{dy}{dx} \)

Slope of normal = \( -\frac{1}{\frac{dy}{dx}} = -\frac{dx}{dy} \)

Tangent: \( y - y_1 = \frac{dy}{dx}(x - x_1) \)

Normal: \( y - y_1 = -\frac{1}{\frac{dy}{dx}}(x - x_1) \)

Length of tangent = \( \left| \frac{y \sqrt{1 + (y')^2}}{y'} \right| \)

Length of normal = \( \left| y \sqrt{1 + (y')^2} \right| \)

Length of subtangent = \( \left| \frac{y}{y'} \right| \)

Length of subnormal = \( \left| y y' \right| \)

Rolle's Theorem: If \( f(a) = f(b) \), then \( f'(c) = 0 \) for some \( c \in (a,b) \)

Lagrange's MVT: \( f'(c) = \frac{f(b) - f(a)}{b - a} \)