AP Calculus BC Unit 9 – Parametric Equations, Polar Coordinates, and Vector-Valued Functions

A curve in the plane is defined parametrically by the equations x =  2sin(1+3t) and y= 2t³   .  Find dy/dx

A curve in the plane is defined parametrically by the equations   and . Find the value of   at  .

A curve is defined by the parametric equations x=3 e^2t  and y=e ^3t −1 .  What is    in terms of t ?

A curve is defined by the parametric equations x=8 √ t+1 and  y=−6 √ t+t . What is    in terms of t ?

Consider the parametric curve: ,

Which integral gives the arc length of the curve over the interval from  to ?

Consider the parametric curve:   x= 5 t ² , y=2 e ^t   Which integral gives the arc length of the curve over the interval from t =−1 to t =3 ?

Let f be a vector­valued function defined by    Find f'(t) ;

Let g be a vector­valued function defined by    Find g'(t)….

Let h be a vector­valued function defined by    Find h’s second derivative h′′(t).

A particle moves in the xy­plane so that at any time t ≥ 0 its coordinates are x= t³ – 2t and y= 3t+1  What is the particle’s velocity vector at t=3 ?

Let r be the polar function r ( θ )=5 θ−1. What is the rate of change of the y­coordinate with respect to θ at the point  where θ=π ?

Let r be the polar function Here is its graph for 0 ≤θ ≤ 2 π : What is the rate of change of the x-coordinate with respect to θ at the point P?

Consider the polar curve r=4 sin ⁡ ( 5 θ) . What is the equation of the tangent line to the curve r at 

Let R be the region enclosed by the polar curve    Which integral represents the area of R?

Let R be the region enclosed by the polar curve    Which integral represents the area of R?

Let R be the region in the second quadrant enclosed by the polar curve r ( θ )=θ+ sin ⁡ ( θ) and the coordinate axes, as shown in the graph. Which integral represents the area of R?

Let R be the region that is inside the polar curve  r = 3 and outside the polar curve r=2+ sin ( θ ) , as shown in the graph. Which integral represents the area of R?

Let R be the region in the second quadrant that is inside the polar curve r= 2 and  outside the polar curve r=2+ sin ⁡ ( 2 θ) , as shown in the graph.  Which integral represents the area of R?

Let R be the region inside the polar curve r=cos ⁡(θ) and inside the polar curve r=1+ sin ⁡ ( θ) , as shown in the graph. Which integral represents the area of R?