AP Calculus BC Parametrics, Vector & Polar Formulas

For differentiable parametric functions,  $y\left(t\right)\ \&\ x\left(t\right)$ , the first derivative  $y'\left(x\right)=\frac{\text{d}y}{\text{d}x}$  is

•  $\frac{\text{d}x}{\text{d}t}$  

•  $\frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}\ where\ \frac{\text{d}x}{\text{d}t}\ne0$  

•  $\frac{\text{d}y}{\text{d}t}$  

•  $\frac{\frac{\text{d}x}{\text{d}t}}{\frac{\text{d}y}{\text{d}t}}where\ \frac{\text{d}y}{\text{d}t}\ne0$  

For differentiable parametric functions,  $y\left(t\right)\ \&\ x\left(t\right)$ , the second derivative  $\frac{\text{d}^2y}{\text{d}x^2}$  is

•  $\frac{\text{d}\left(\frac{dy}{dt}\right)}{\text{d}t}$  

•  $\frac{\frac{\text{d}y}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}\ where\ \frac{\text{d}x}{\text{d}t}\ne0$  

•  $\frac{\text{d}^2y}{\text{d}t^2}$  

•  $\frac{\frac{\text{d}y'}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}where\ y'=\frac{\text{d}y}{\text{d}x}\ and\ \frac{\text{d}x}{\text{d}t}\ne0$  

The length of a parametric curve defined by continuous and differentiable functions  $y\left(t\right)\ and\ x\left(t\right)$ , on the interval  $a\le t\le b$ , is: 

•  $L=\sqrt{x\left(t\right)^2+y\left(t\right)^2}$  
  

•  $L=\sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2+\left(\frac{\text{d}y}{\text{dt}}\right)^2}$  
  

•  $L=\int_a^b\sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2+\left(\frac{\text{d}y}{\text{dt}}\right)^2}dt$  

•  $L=\int_a^b\sqrt{x\left(t\right)^2+y\left(\text{t}\right)^2}dt$  

The magnitude, or absolute value, of the vector   $<x,\ y>$ is: 

•  $\left|<x,y>\right|=\sqrt{x^2+y^2}$  
  

•  $\left|<x,y>\right|\sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2+\left(\frac{\text{d}y}{\text{dt}}\right)^2}$  
  

•  $\left|<x,y>\right|=\int_a^b\sqrt{\left(x'\left(t\right)\right)^2+\left(y'\left(t\right)\right)^2}dt$  

•  $\left|<x,y>\right|\int_a^b\sqrt{x\left(t\right)^2+y\left(\text{t}\right)^2}dt$  

Tags

CCSS.HSN.VM.A.1

 $\overrightarrow{r}\left(t\right)=<x\left(t\right),y\left(t\right)>$  , for differentiable  $x\left(t\right),\ \ y\left(t\right)$  

• position vector r(t)

• velocity vector r(t)

• velocity vector v(t)

• acceleration vector a(t)

 $\overrightarrow{v}\left(t\right)=<\frac{\text{d}x}{\text{d}t},\frac{\text{d}y}{\text{d}t}>$  , for differentiable  $x\left(t\right),\ \ y\left(t\right)$  

• position vector r(t)

• position vector v(t)

• velocity vector v(t)

• acceleration vector a(t)

 $\overrightarrow{a}\left(t\right)=<\frac{\text{d}^2x}{\text{d}t^2},\frac{\text{d}^2y}{\text{d}t^2}>$  , for differentiable  $x\left(t\right),\ \ y\left(t\right)$  

• position vector a(t)

• velocity vector a(t)

• velocity vector v(t)

• acceleration vector a(t)

The speed of the vector for differentiable  $x\left(t\right),\ y\left(t\right)$  is: 

•  $\sqrt{x^2+y^2}$  
  

•  $\sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2+\left(\frac{\text{d}y}{\text{dt}}\right)^2}$  
  

•  $\int_a^b\sqrt{\left(x'\left(t\right)\right)^2+\left(y'\left(t\right)\right)^2}dt$  

•  $\int_a^b\sqrt{x\left(t\right)^2+y\left(\text{t}\right)^2}dt$  

The direction of motion or the direction vector is:

• $\frac{\left|v\right|}{\overrightarrow{v}}$

• $\overrightarrow{\frac{v}{\left|v\right|}}$

• $\sqrt{x^2+y^2}$

• $\sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2+\left(\frac{\text{d}y}{\text{d}t}\right)^2}$

The displacement for $\overrightarrow{v}\left(t\right)=<v_x\left(t\right),v_y\left(t\right)>$  for  $t=a\ to\ b$  is:

•  $<\int_a^bv_x\left(t\right)dt,\ \int_a^bv_y\left(t\right)dt>$  

•  $<\int_a^ba_x\left(t\right)dt,\ \int_a^ba_y\left(t\right)dt>$  

•  $<\int_a^br_x\left(t\right)dt,\ \int_a^br_y\left(t\right)dt>$  

•  $<\int_a^b\left|v_x\left(t\right)\right|dt,\ \int_a^b\left|v_y\left(t\right)\right|dt>$