(a) Use R to plot the heart-shape curve defined by the following parametric equations

x= 4 sin(2t) + 9 sin(t)

y = 4 cos(2t) + 9 cos(t)

for 0 ≤t≤ 2r. Please include both the figure and R. code in your solution pdf file.

(b) Find the velocity function v(t) = d = (dd) for a point moving along this curve, where the r= (x, y) may be regarded as a position vector. You can choose to find derivatives by hand or by R. You are also allowed to use R to verify your answer.

(c) Find the acceleration function a(t) = dx = (d) for a point moving along this curve.

(d) Based on the results from Parts (b) and (c), evaluate the vector of v, and a at time t = 1.5.

Use R to compute values, e.g., t= 1.5; 2*cos (3*t) - 6*t*sin (3*t) #= 8.376179.

(e) Evaluate the speed at t = 2.5, i.e., the vector length |v| where v is from Part (b) at t = 2.3.

(f) Evaluate the magnitude of the acceleration vector at t= 2.3, i.e., the vector length a where a is from Part (c) at t = 2.3.