I need some help with math

Question

For 0 ≤ t ≤ 12 , a roller coaster car has position (x(t),y(t)) with velocity vector ⟨10, −tsin(πt/4)⟩ , where x(t) and y(t) are measured in meters and t is measured in seconds. At time t=0 , the car is at position (0,18) .

a) What is the slope of the line tangent to the path of the car at time t=2?

b) What is the speed of the car at time t=6 ?

c) During the time interval 0 ≤ t ≤ 12 , what is the maximum value of y(t)?

d) Write, but do not evaluate, an expression that gives the average speed of the car over the time interval 0 ≤ t ≤ 12.

Answer 1

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For a) want dy/dx at t=2 and we can use dy/d/t/dx/dt= -2/10

b) at t=6, v_x=10, v_y=6 and |v|=√136

c) we have to be careful with d²y/dx² in parametric equations, but we can still examine

d²y/dt² and find at t=8 the value is negative meaning y(t) has a local maximum at t=8.

Either the maximum y is 18 at t=0 or it is at t=8

for d) the average velocity is change in position divided by change in time

and the change in position is √((change in x)²+(change in y)²)

change in x is 120 meters,

change in y is (the integral from 0 to 12 of -t*sin(πt/4))

and change in time is 12