Thread: 2 Calculus questions... (particle motion and derivatives)

1) A particle moves along the x-axis so that, at any time t≥0, its acceleration is given by a(t)=6t-3. At time t=0, the velocity of the particle is -18, and its position is -27.
a) Find v(t), the velocity of the particle at any time t≥0.
b) For what values of t≥0 is the particle moving to the right?
c) Find x(t), the position of the particle at any time t≥0.

2) Let y=2e^(sinx)
a) Calculate dy/dx (1st derivative) and d2y/dx2 (2nd derivative).
b) If x and y both vary with time in such a way that y increased at a steady rate of 5 units per second, at what rate is x changing when x=pi?

Any help would be appreciated, even if you just explain to me the steps or methods I should be using.

Originally Posted by macroecon

1) A particle moves along the x-axis so that, at any time t≥0, its acceleration is given by a(t)=6t-3. At time t=0, the velocity of the particle is -18, and its position is -27.
a) Find v(t), the velocity of the particle at any time t≥0.
b) For what values of t≥0 is the particle moving to the right?
c) Find x(t), the position of the particle at any time t≥0.

2) Let y=2e^(sinx)
a) Calculate dy/dx (1st derivative) and d2y/dx2 (2nd derivative).
b) If x and y both vary with time in such a way that y increased at a steady rate of 5 units per second, at what rate is x changing when x=pi?

Any help would be appreciated, even if you just explain to me the steps or methods I should be using.

for (a) i think you should integrate a(t).
for (c) the same thing, integrate v(t).

so for #1
a) i ended up with v(t)=3t^2-3t-18 or 3(t-3)(t+2)
b) t≥3
c) x(t)=(t^3)-(3/2t^2)-18t-27

Can anyone check that?

Originally Posted by macroecon

so for #1
a) i ended up with v(t)=3t^2-3t-18 or 3(t-3)(t+2)
b) t≥3
c) x(t)=(t^3)-(3/2t^2)-18t-27

Can anyone check that?

I agree .