Differentiable Parametric Equations
Theorem: Let C be a curve with parametric equations x = x(t) and y = y(t) which are differentiable functions. If there exists a differentiable function f such that y(t) = f(x(t)) for t in some open interval, then


>
Corollary: Let C be a curve with parametric equations x = x(t) and y = y(t) which are twice differentiable functions. If there exists a twice differentiable function f such that y(t) = f(x(t)) for t in some open interval, then


Example:
>

To find the equation of the tangent line of



First, we find the derivative:



The equation of the tangent line is:

---

Vectors and their Geometry
Definitions.
A two-dimensional vector is an ordered pair of real numbers v = (a, b).
a and b are called the components of v.
The length or magnitude of the vector v = (a, b) is



>>