Calc unit 2

Average Rate of Change

Has 2 points, The slope of the secant line from x=a to x=a+h.

Instantaneous Rate of Change

Has 1 point. The slope of the Tangent line at x=a.

Equation of the Tangent line- Taylors form:

y = y₁ + m ( x - x₁ )

Differentiability

We can find the derivative. Must be a smooth line or curve. The derivative fails to exist where the function has: Discontinuity, Vertical Tangent line, and sharp turns.

Normal Lines

Perpendicular to the Tangent line. Opposite and reciprocal slope

\frac{d}{dx}a^{x}=

a^{x}\cdot\ln a

\frac{d}{dx}\exponentialE^{x}=

\exponentialE^{x}