Consider the curve in the xy plane represented by $$x=e^t$$  and  $$y=te^{-t}$$  for  $$t\ge0$$  .  The slope of the tangent to the curve at the point when x=3 (not t=3) is...

The flight of a paper airplane is modeled by the curve given by $x=t-3\sin t$ and  $y=4-3\cos t$ for time t in seconds and  $0\le t\le10$ (x and y are in feet). What is the distance traveled by the paper airplane?

The flight of a paper airplane is modeled by the curve given by $x=t-3\sin t$ and  $y=4-3\cos t$ for time t in seconds and  $0\le t\le10$ (x and y are in feet). What describes the motion of the paper airplane at t=7 seconds?

The flight of a paper airplane is modeled by the curve given by $x=t-3\sin t$ and  $y=4-3\cos t$ for time t in seconds and  $0\le t\le10$ (x and y are in feet). What is the slope of the tangent line to the path of the paper airplane at t=7 seconds?

Consider the curve given by $x=t^3+2t-1$  and  $y=t^2-t+5$ .  What is the arc length of the curve for  $0\le t\le2$  ? 

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

Find $\frac{dy}{dx}$ for the curve given by $x=\left(t-1\right)^3$ and $y=\sqrt{t}$ . 

The length of the curve determined by $x=t^2$ and  $y=t$  from t=0 to t=4 is

Find the equation of the tangent line for the curve defined by $x=2t$ and $y=t^2+5$ at the point where t=2. 

Consider the curve given by $x=2\sin t$ and  $y=4\cos^2t$ for  $0\le t\le\pi$ .  Find  $\frac{d^2y}{dx^2}$  at  $t=\frac{\pi}{4}$  

Find the slope at the value t=3 for the parametric equations $x=t+10$ and  $y=t^2+2t$  

Find the concavity at the value t=3 for the parametric equations $x=t+10$ and  $y=t^2+2t$  

Find all points of horizontal tangency to the curve $$x=7+3\cos t$$  and  $$y=-5+\sin t$$  

If $$r\left(t\right)=<5t^2-2t,3-\ln t>$$ , then  $$-\frac{1}{3}r'\left(2\right)=$$   

Given $$r'\left(t\right)=<te^{-t^2},-e^{-t}>$$ and  $$r\left(0\right)=<\frac{1}{2},-1>$$ find r(t) that satisfies the initial condition.  

If $$r\left(t\right)=<e^{-4t}+9,7-4t^3>$$ , then r"(t)= 

If $$\frac{dR}{dt}=<\frac{3}{2}\sqrt{t+1},e^{-t}>$$ where $$R\left(0\right)=<0,0>$$  , find $$R\left(t\right)$$ . 

$$\int_0^2<e^t,-te^t>dt=$$  

For $t\ge0$ , a particle moves along a curve so that its position at time t is given by (x(t),y(t)).  At t=3, the particle is at position (6,2).  Given $\frac{dx}{dt}=1-\cos^2t$ and  $\frac{dy}{dt}=\frac{t^{1.5}}{1.2^t}$ , is the vertical movement of the particle up or down at t=3? 

For $t\ge0$ , a particle moves along a curve so that its position at time t is given by (x(t),y(t)).  At t=3, the particle is at position (6,2).  Given $\frac{dx}{dt}=1-\cos^2t$ and  $\frac{dy}{dt}=\frac{t^{1.5}}{1.2^t}$ , find the y-coordinate of the particle's position at t=5

For $t\ge0$ , a particle moves along a curve so that its position at time t is given by (x(t),y(t)).  At t=3, the particle is at position (6,2).  Given $\frac{dx}{dt}=1-\cos^2t$ and  $\frac{dy}{dt}=\frac{t^{1.5}}{1.2^t}$ , find the speed of the particle at t=5.

For $t\ge0$ , a particle moves along a curve so that its position at time t is given by (x(t),y(t)).  At t=3, the particle is at position (6,2).  Given $\frac{dx}{dt}=1-\cos^2t$ and  $\frac{dy}{dt}=\frac{t^{1.5}}{1.2^t}$ , find the distance traveled by the particle from time t=3 to t=5.

The position of a particle moving in the xy-plane is given by (x(t),y(t)) for which $x'\left(t\right)=t\sin t$ and $y'\left(t\right)=5e^{-3t}+2$ .  What is the slope of the line tangent to the path of the particle at the point at which t=2? 

The position of a particle moving in the xy-plane is given by $x\left(t\right)=t^3-3t^2$  and  $y\left(t\right)=12t-3t^2$ .  At which point is the particle at rest? 

The position of a particle moving along a curve at time

For $$t\ge0$$ , a particle is moving along a curve so that its position at time t is given by the parametric equations  $$x\left(t\right)=\sin t$$  and  $$y\left(t\right)=t^2-t+3$$  .  Which of the following expressions gives the speed of the particle at time t? 

At time t, a particle moving along a curve in the xy-plane has position (x(t), y(t)) where $$x\left(t\right)=\sin\left(\frac{\pi t}{3}\right)$$  and  $$y\left(t\right)=t^2-5$$  .  Which of the following gives the direction of motion for the particle at t=2?

For $$t\ge0$$ , a particle is moving along a curve so that its position at time t is given by the parametric relations x(t) (the graph) and  $$y\left(t\right)=t^2-3t+1$$ .  Which of the following expressions gives the total distance the particle travels on the interval [0,4]

A curve is defined parametrically by the equations $$x\left(t\right)=3\sqrt{t}$$ and $$y\left(t\right)=\frac{1}{4}t^2$$ .  Which of the following represents the equation of the line tangent to the curve at t=4? 

For what value(s) of t does the curve defined by the parametric equations $$x=\frac{4}{t^3-2t^2-4}$$ and  $$y=\frac{2}{t}+\frac{1}{8}t^2$$ have a horizontal tangent?  

The position of a particle moving in the xy-plane is given by the parametric equations $$x\left(t\right)=2^{\cos t}$$ and $$y\left(t\right)=2^{\sin t}$$  for time $$t\ge0$$ .  What is the speed of the particle when t=1.2?  

At $$t\ge0$$ , a particle moving in the xy-plane has a position vector given by $$r\left(t\right)=<t^2-t+4,e^{2t-10}>$$ .  What is the speed of the particle at time t=5? 

An object moves in the xy-plane so that its position at any time t is given by the parametric equations $$x\left(t\right)=e^{2t}$$ and $$y\left(t\right)=2t^3-t^2+4$$ .  What is the slope of the tangent line at t=2? 

Find the arc length of the curve defined by $$x=\arcsin t,\ y=\ln\sqrt{1-t^2}$$  on the interval  $$0\le t\le\frac{1}{2}$$  

Find dy/dx for the graph of the parametric equations

Find  $$\frac{d^2y}{dx^2}$$  for the graph of the parametric equations

$$x=t^2,y=t^2+6t+5$$  

Find  $$\frac{dy}{dx}$$  for the graph of the parametric equations

Find  $$\frac{d^2y}{dx^2}$$  for the graph of the parametric equations

Find  $$\frac{dy}{dx}$$  for the graph of the parametric equations

Find  $$\frac{d^2y}{dx^2}$$  for the graph of the parametric equations

For the curve defined by the parametric equations $$x=2\cos t,\ y=3\sin t$$ , find the tangent line where  $$t=\frac{\pi}{4}$$  

Given the parametric equations $$x=t^2-t+1,\ y=t^3-3t$$ , find the point(s) where the curve has a horizontal tangent. 

Given the parametric equations $$x=t^2-t+1,\ y=t^3-3t$$ , find the point(s) where the curve has a vertical tangent. 

Given the parametric equations $$x=3+2\cos t,\ y=-1+4\sin t$$ , find the point(s) where the curve has a vertical tangent. 

What is the arc length of the curve created by the parametric equations $$x=t^2,\ y=t^3,\ 0\le t\le2$$  

What is the arc length of the curve created by the parametric equations $$x=e^{2t}+1,\ y=3t-1,\ -2\le t\le2$$  

Identify the curve of $$x=2t+1,\ y=t^2+2$$ .

At time t on [0,2], the velocity of a particle moving along the x-axis is given by $$v\left(t\right)=e^{t^2}-2$$ .  What is the total distance traveled by the particle during the time interval [0,2]? 

The velocity of a particle moving in a straight line for $$t\ge0$$  is given by  $$v\left(t\right)=\ln\left(t^3+1\right)$$  .  What is the acceleration of the particle at t=4?

At time $$t\ge0$$ , a particle moving in the xy-plane has velocity vector given by $$v\left(t\right)=<4e^{-t},\sin\left(1+\sqrt{t}\right)>$$ .  What is the total distance the particle travels between t=1 and t=3? 

For time $$t\ge0$$ seconds, the position of an object traveling along a curve in the xy-plane is given by the parametric equations x(t) and y(t), where $$\frac{dx}{dt}=t^2+3$$ and $$\frac{dy}{dt}=t^3+t$$ .  At what time t is the speed of the object 10 units per second? 

At time t>0, the position of a particle moving along a curve in the xy-plane is (x(t),y(t)) where $$\frac{dx}{dt}=t-5\cos t$$ and $$\frac{dy}{dt}=6\cos\left(1+\sin t\right)$$ .  At time t=3, the particle is at position (-1,2). What is the equation of the tangent line to the path of the particle at t=3? 

At time t>0, the position of a particle moving along a curve in the xy-plane is (x(t),y(t)) where $$\frac{dx}{dt}=t-5\cos t$$ and $$\frac{dy}{dt}=6\cos\left(1+\sin t\right)$$ .  At time t=3, the particle is at position (-1,2).

Find the time where the path of the particle is vertical.  Is the motion up or down at that moment?

At time t>0, the position of a particle moving along a curve in the xy-plane is (x(t),y(t)) where $$\frac{dx}{dt}=t-5\cos t$$ and $$\frac{dy}{dt}=6\cos\left(1+\sin t\right)$$ .  At time t=3, the particle is at position (-1,2). Find the y-coordinate of the particle's position at t=0.

At time t>0, the position of a particle moving along a curve in the xy-plane is (x(t),y(t)) where $$\frac{dx}{dt}=t-5\cos t$$  and $$\frac{dy}{dt}=6\cos\left(1+\sin t\right)$$ .   At time t=3, the particle is at position (-1,2).

Find the total distance traveled by the particle for [0,3].

For t>0, a particle is moving along a curve so that its position at time t is ((x(t),y(t)).  At time t=2, the particle is at position (1,5).  It is known that $$\frac{dx}{dt}=\frac{\sqrt{t+2}}{e^t}$$ and $$\frac{dy}{dt}=\sin^2t$$ .  Find the x-coordinate of the particle's position at time t=4.

For t>0, a particle is moving along a curve so that its position at time t is ((x(t),y(t)).  At time t=2, the particle is at position (1,5).  It is known that $$\frac{dx}{dt}=\frac{\sqrt{t+2}}{e^t}$$ and $$\frac{dy}{dt}=\sin^2t$$ .  Find the speed of the particle at t=4.

For t>0, a particle is moving along a curve so that its position at time t is ((x(t),y(t)).  At time t=2, the particle is at position (1,5).  It is known that $$\frac{dx}{dt}=\frac{\sqrt{t+2}}{e^t}$$ and $$\frac{dy}{dt}=\sin^2t$$ .  Find the y component of the acceleration vector at t=4.