A car is driving along a straight road and its position (in meters) at time t (in seconds) is given by the equation: s(t) = 3t^2 + 12t + 5. Find the car's velocity at time t=3 seconds, using calculus.
user
Sure, I'd be happy to help with this calculation.

The formula for velocity is v = Δs / Δt, where v is velocity, Δs is the change in position, and Δt is the change in time. In this case, we're given the equation s(t) = 3t^2 + 12t + 5, which is the position at time t, so we need to find the Δs for a small change in time Δt = 3 - t = Δt.

So, let's calculate the change in position from time t to t + Δt = t + 3 = t + Δt:

Δs = s(t + Δt) - s(t) = [3(t + Δt)^2 + 12(t + Δt) + 5] - [3t^2 + 12t + 5]

Simplifying this expression and rearranging:

∆s = 6t + 36 + 12t + 6Δt + 12tΔt + 12Δt^2

= 18t + 36 + 12t = 36t + 36

So, Δs = (36 * 3) + 36 = 126 meters (since Δt = 3 seconds).

Now, we can plug this into the velocity formula:

v = Δs / Δt = 126 / 3 = 42 meters per second.

So, the velocity (or speed) of the car at time t=3 seconds, based on this linear motion model, is approximately 42 meters per second. This calculation assumes that the car's motion in the given time period (three seconds) is linear. If the motion changes during this time period, a more complex calculus approach would be required.
assistant
A car is driving along a straight road and its position (in meters) at time t (in seconds) is given by the equation: s(t) = 3t^2 + 12t + 5. Find the car's velocity at time t=3 seconds, using calculus.
user
To find the car's velocity at time t = 3 seconds, we need to take the derivative of the position function s(t) with respect to time t. The derivative of a function represents the rate of change of that function, which in this case is the velocity.

Given s(t) = 3t^2 + 12t + 5, let's find its derivative:

s'(t) = d/dt [3t^2 + 12t + 5]

Using basic differentiation rules:

s'(t) = 6t + 12

Now that we have the velocity function s'(t), we can plug in t = 3 to find the car's velocity at that time:

Velocity at t=3: s'(3) = 6(3) + 12

Velocity at t=3: s'(3) = 18 + 12

Velocity at t=3: s'(3) = 30

So the car's velocity at time t = 3 seconds is 30 meters per second.
assistant