I am not going to just do these for you. Did you try? Where did you get stuck?
Plug and chug, for example:
ds/dt = t* d/dt(ln 2t) + ln 2t
= t * (1/2t)2 + ln 2 t
= 1 + ln 2t
when is that zero?
ln 2t = -1
e^ln 2t = 2t =1/e
t = 1/(2e)
a = d^2s/dt^2 = d/dt(ds/dt)
= 0 + d/dt(ln2t) = (1/2t)2 = 1/t
so when t = 1/2e
that acceleration is 2e
1) A particle is moving along the x-axis so that its position at t ≥ 0 is given by s(t) = (t)In(2t). Find the acceleration of the particle when the velocity is first zero.
2) The driver of a car traveling at 50 ft/sec suddenly applies the brakes. The position of the car is s(t) = 50t - 2t^2, t seconds after the driver applies the brakes. How many seconds after the driver applies the brakes does the car come to a stop?
3) The position of a particle on the x-axis at time t, t > 0, is s(t) = e^t with t measured in seconds and s(t) measured in feet. What is the average velocity of the particle for 0 ≤ t ≤ In3?
4) A particle moves along the x-axis so that at any time t, measured in seconds, its position is given by s(t) = sin(t) - 4cos(2t), measured in feet. What is the acceleration of the particle at time t = π seconds?
5) A particle moves with velocity function v(t) = -t^2 + 5t - 3, with v measured in feet per second and t measured in seconds. Find the acceleration of the particle at time t = 3 seconds.
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