Calculus II - Parametric equations

8 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Parametric equation

   Suppose x=x(t)x=x(t)x=x(t) and y=y(t)y=y(t)y=y(t) are two functions of a third variable, ttt , called the parameter, that are defined on the same interval III . Then the equations x=x(t)x=x(t)x=x(t) and y=y(t)y=y(t)y=y(t) where ttt is in III are called parametric equations, also represented through the form (x,y)=(x(t),y(t))(x,y)=(x(t),y(t))(x,y)=(x(t),y(t))

2. Plane curve

   The graph of the points defined by a parametric equation

3. Finding a rectangular equation from parametric equations

   Solve for the parameter in one of the equations, then substitute this value for the parameter as the parameter in the other equation

4. Curvilinear motion

   Motion as described by parametric equations dependent on time

5. Parametric equations of a cycloid

   x(t)=a(t−sin⁡t)x(t)=a(t-\sin{t})x(t)=a(t−sint) y(t)=a(1−cos⁡t)y(t)=a(1-\cos{t})y(t)=a(1−cost) −∞<t<∞-\infty < t < \infty−∞<t<∞

6. Smooth (parametric) curve

   A plane curve CCC of parametric equations x=x(t)x=x(t)x=x(t) , y=y(t)y=y(t)y=y(t) is smooth both functions are continuous over the closed interval [a,b][a,b][a,b] , differentiable on the open interval (a,b)(a,b)(a,b) , and have derivatives dxdt\frac{dx}{dt}dtdx​ and dydt\frac{dy}{dt}dtdy​ that are both continuous and never simultaneously 0 on (a,b)(a,b)(a,b)

7. Slope of a tangent line to a smooth (parametric) curve

   dydx=dydtdxdt\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}dxdy​=dtdx​dtdy​​ , provided dxdt≠0\frac{dx}{dt} \neq 0dtdx​=0 When dxdt=0\frac{dx}{dt}=0dtdx​=0 but dydt≠0\frac{dy}{dt} \neq 0dtdy​=0 , there is a vertical tangent line When dydt=0\frac{dy}{dt}=0dtdy​=0 but dxdt≠0\frac{dx}{dt} \neq 0dtdx​=0 , there is a horizontal tangent line

8. Arc length for parametric equations

   For a smooth curve CCC of parametric equations x=x(t)x=x(t)x=x(t) , y=y(t)y=y(t)y=y(t) , the arc length sss from t=at=at=a to t=bt=bt=b is s=∫ab(dxdt)2+(dydt)2dts=\int_a^b\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dts=∫ab​(dtdx​)2+(dtdy​)2 ​dt