Question

The position of a particle moving along the x-axis is given by s(t)=3t 2 +3. Use difference quotients to find the velocity v(t) and acceleration a(t), filling in the following expressions as you do so: v(t)=lim h→0​ [a(t)=lim h→0​ [​ /h]=/h]=​

Answer 1

The velocity of the particle is v(t) = 6t, found by using the difference quotient and limit process. The acceleration a(t) is the derivative of velocity, which is a constant 6 m/s^2.

Step-by-step explanation:

The student's question involves finding the velocity v(t) and acceleration a(t) of a particle whose position as a function of time is given by s(t) = 3t2 + 3. To find the velocity, we use the difference quotient and take the limit as h approaches zero:

v(t) = limh→0 ​[s(t + h) - s(t)] / h

To calculate this, we substitute s(t):

v(t) = limh→0 ​[(3(t + h)2 + 3) - (3t2 + 3)] / h = limh→0 ​[3h2 + 6th] / h = limh→0 ​[3h + 6t]

After canceling h and applying the limit, we get:

v(t) = 6t

Next, we find the acceleration by taking the derivative of velocity:

a(t) = dv(t)/dt = d(6t)/dt = 6

Therefore, the acceleration is a constant 6 m/s2.