Parametric Equations

If  $x=e^{-t}$  and  $y=\cos\left(2t\right)$  then $\frac{dy}{dx}=$  

•  $\frac{e^{-t}}{2\sin\left(2t\right)}$  

•  $\frac{2\sin\left(2t\right)}{e^{-t}}$  

•  $\frac{-e^{-t}}{2\sin\left(2t\right)}$  

•  $\frac{-2\sin\left(2t\right)}{e^{-t}}$  

A particle moves on a plane curve so that at any time t > 0 its x-coordinate is  $t^3-t$  and its y-coordinate is  $\left(2t-1\right)^3$  . The acceleration vector of the particle at t = 1 is

• (2,3)

• (2,6)

• (6,12)

• (6,24)

Tags

CCSS.HSF.IF.A.2

CCSS.HSN.VM.A.1

CCSS.HSN.VM.A.3

The length of the path described by the parametric equations  $x=\frac{1}{3}t^3$  and  $y=\frac{1}{2}t^2$  where  $0\le t\le1$  is given by 

•  $\int_0^1\sqrt{t^2+1}dt$  

•  $\int_0^1\sqrt{t^2+t}dt$  

•  $\int_0^1\sqrt{t^4+t^2}dt$  

•  $\frac{1}{2}\int_0^1\sqrt{4+t^4}dt$  

A curve C is defined by the parametric equations  $x=t^2-4t+1$  and  $y=t^3$  . Which of the following is an equation of the line tangent to the graph of C at the point (-3,8)?

•  $x=-3$  

•  $y=8$  

•  $y=-\frac{27}{10}\left(x+3\right)+8$  

•  $y=12\left(x+3\right)+8$  

Tags

CCSS.HSA.REI.C.7

CCSS.HSG.GPE.B.5

Find  $\frac{d^2y}{dx^2}$  for the curve given by  $x=t^2+1$  and  $y=t^3$  .

•  $\frac{3}{4t}$  

•  $\frac{3}{2t}$  

•  $3t$  

•  $\frac{3}{2}$  

If  $x=\cos t$  and  $y=2\sin^2t$  then  $\frac{dy}{dx}$  at t = 1 is

• -2cos1

• -4cos1

• -2tan1

• -2sin1

For what value(s) of t does the curve defined by the parametric equations  $x=\frac{5t}{1+t^3}$  and  $y=\frac{2t^2}{1+t^3}$  have a horizontal tangent?

• 0 only

•  $\ ^3\sqrt{2}$  only

• 0 and 4 only 

• 0 and  $\ ^3\sqrt{2}$  only 

An object moving along a curve in the xy-plane is in position  $\left(x\left(t\right),y\left(t\right)\right)$  at time  $t\ge0$  with  $\frac{dx}{dt}=2-\sin\left(t^2\right)$  . At time t = 3, the object is at position (2,7). What is the x coordinate of the position of the object at time t = 6? 

• 8.135

• 9.762

• 10.375

• 11.308

Tags

CCSS.HSA.CED.A.2

A curve in the plane is defined parametrically by the equations  $x=t^3+t$  and  $y=t^4+2t^2$  . An equation of the line tangent to the curve at t = 1 is 

• y = 2x - 1

• y = 4x - 5

• y = 8x

• y = 8x + 13

Tags

CCSS.HSA.CED.A.2

Eliminate the parameter.  $x=\frac{1}{t-2}$  and  $y=4t-5$  

•  $y=4x+5$  

•  $y=\frac{4}{x}+3$  

•  $y=-2x+3$  

•  $y=4x+\frac{1}{3}$  

Tags

CCSS.HSA.CED.A.2

CCSS.HSA.SSE.B.3