9.1 Defining and Differentiating Parametric Equations

    Cards (13)

    • In parametric equations, xx and yy are expressed as functions of the same parameter tt.
    • What do f(t)f(t) and g(t)g(t) represent in parametric equations?

      Functions of tt
    • As tt changes in the equations \begin{cases} x =ty= t \\ y = t^{2} \end{cases}, the point moves along a parabola y=y =x2 x^{2} called its path
    • Parametric equations allow for easy representation of complex shapes that are difficult to express as explicit functions.
    • Match the curve with its parametric equations:
      Circle ↔️ x=x =rcos(t),y= r\cos(t), y =rsin(t) r\sin(t)
      Ellipse ↔️ x=x =acos(t),y= a\cos(t), y =bsin(t) b\sin(t)
      Line ↔️ x=x =x0+ x_{0} +at,y= at, y =y0+ y_{0} +bt bt
    • Order the steps to describe the movement of a point along a curve defined by parametric equations as tt changes:

      1️⃣ Change the value of tt
      2️⃣ Calculate the new coordinates (x,y)(x, y)
      3️⃣ Plot the point in the plane
      4️⃣ Repeat the process
    • To find dydx\frac{dy}{dx} from parametric equations, we use the formula dydx=\frac{dy}{dx} =dy/dtdx/dt \frac{dy / dt}{dx / dt}, which is called the chain rule
    • The parametric derivative dydx\frac{dy}{dx} gives the slope of the tangent line to the curve at a specific point.
    • The equation of the tangent line to a parametric curve at (x_{0}, y_{0})</latex> is yy0=y - y_{0} =dydx(t0)(xx0) \frac{dy}{dx}(t_{0}) (x - x_{0}), where dydx(t0)\frac{dy}{dx}(t_{0}) is the slope at the parameter value t_{0}
    • The velocity vector in parametric equations is given by v=\vec{v} =(dxdt,dydt) \left( \frac{dx}{dt}, \frac{dy}{dt} \right).
    • Steps to find the tangent line to the curve defined by x=x =t2 t^{2} and y=y =2t 2t at t=t =1 1:

      1️⃣ Find the derivatives dxdt\frac{dx}{dt} and dydt\frac{dy}{dt}
      2️⃣ Calculate dydx\frac{dy}{dx}
      3️⃣ Find the coordinates (x0,y0)(x_{0}, y_{0}) at t=t =1 1
      4️⃣ Write the tangent line equation
    • At t=t =1 1, the slope dydx(1)\frac{dy}{dx}(1) for the curve defined by x=x =t2 t^{2} and y=y =2t 2t is 1
    • Parametric equations define the coordinates of a point in terms of a parameter t
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