Please do all parts/questions!! thank you! Eliminate the parameter to find a des

Question

Please do all parts/questions!! thank you!

Eliminate the parameter to find a description of the following circles or circular arcs in terms of x and y. Give the center and radius, and indicate the positive orientation. x = 4 cos t, y = 3 sin t; 0 lessthanorequalto t lessthanorequalto 2 pi Determine dy/dx in terms of t and evaluate it at the given value of t for the following parametric curves. x = 3 sin t, y = 2 cos t; t = pi/4 x = 3t, y = 2t^3; t = 1 Graph the point with the following polar coordinates. Give two alternative representations of the point in polar coordinates. (-4, pi/6)

Explanation / Answer

1)x=4cost , y =3sint

x/4=cost ,y/3=sint

cos2t +sin2t =1

(x/4)2+(y/3)2=1

(x2/16)+(y2/9)=1

curve is an ellipse with length of semi major axis 4 , length of semi minor axis 3

2) x =3sint , y =2cost

dx/dt =3cost , dy/dt =-2sint

dy/dx =(dy/dt)/(dx/dt)

dy/dx =(-2sint)/(3cost)

dy/dx =(-2/3)tant

at t =pi/4

dy/dx =(-2/3)tan(pi/4)

dy/dx =(-2/3)*1

dy/dx =(-2/3)

b) x=3t , y =2t3

dx/dt =3 ,dy/dt =6t2

dy/dx =(dy/dt)/(dx/dt)

dy/dx =(6t2/3)

dy/dx =2t2

at t= 1

dy/dx =2*12

dy/dx =2

3)(-4,pi/6)

(4,(pi/6)+(pi)) =(4,7pi/6)

(-4,(pi/6)+(2pi)) =(-4,13pi/6) are alternate representations

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