The position function of a moving car is 𝑠 of 𝑡 is equal to six plus five 𝑡 minus four 𝑡 squared for 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 15, where 𝑠 is measured in kilometers and 𝑡 in hours. What is the maximum speed in kilometers per hour of the car on the interval where 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 15?
The question gives us a function for the position of the car at the time 𝑡 where 𝑡 is measured in hours and the position is measured in kilometers, where this position function is only valid on a certain interval for 𝑡. The question wants us to use this to find the maximum speed of our car in kilometers per hour on this interval. To start, we recall that speed is the magnitude of the velocity. And we also recall that the rate of change in the position with respect to time will give us the velocity. So, we can actually use this to find a function for the speed of our car on this interval.
We find the velocity of our car at the time 𝑡 by differentiating the position function with respect to time. That gives us the derivative of six plus five 𝑡 minus four 𝑡 squared with respect to 𝑡. And this is only valid when 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 15. We can differentiate this using the power rule for differentiation. We multiply by the exponent and then reduce the exponent by one. This gives us five minus eight 𝑡.
So, we now have a function for the velocity of our car at the time 𝑡. However, we want to maximize the speed of our car at the time 𝑡. We might be tempted to just write the speed of our car is equal to the absolute value of five minus eight 𝑡 and then try to maximize this function. And this would work. However, there’s actually an easier way. We could actually try to find all of the maxima and the minima of our velocity function five minus eight 𝑡. If we then take the absolute value after we found all of these values, we can then decide which one is the maximum speed.
So, let’s try to find the maxima and minima of our velocity function five minus eight 𝑡. And since 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 15, we only need to consider the values of 𝑡 in the closed interval from zero to 15. The first thing we notice is five minus eight 𝑡 is a polynomial, so it’s continuous for all real numbers. In particular, this means it’s continuous on our closed interval from zero to 15.
And we know how to find the extrema of a continuous function on a closed interval. We do this in the following three steps. First, we find the critical points of that function. And, we recall the critical points of a function are where the derivative is equal to zero or the derivative does not exist. Second, we evaluate the function at any of the critical points. Third, we evaluate the function at the end points of our interval. This will then give us the values of any extrema of our function on that interval.
Let’s apply this to our function 𝑣 of 𝑡 is equal to five minus eight 𝑡. First, let’s find the critical points of our function. Since our function is a polynomial, its derivative is also a polynomial. So, in particular, this means we know it exists on the whole real numbers. So, we won’t find any points where the derivative of our function does not exist.
We now find the turning points of our function. That’s where the derivative is equal to zero. We differentiate five minus eight 𝑡 with respect to 𝑡. And we see that this is just equal to negative eight. And we see that this is never equal to zero. So, there are no turning points of our velocity function.
So, we found all of the critical points. There are no critical points. This means our second step is also complete. We can’t evaluate the function at these points because there weren’t any. This means we only need to evaluate our function at the endpoints of the interval. When 𝑡 is equal to zero, the velocity of our car is five minus eight times zero, which is equal to five. And when 𝑡 is equal to 15, the velocity of our car is five minus eight times 15, which is negative 115.
But remember, we want to find the maximum speed of our car. And speed is the absolute value of the velocity. So, we take the absolute value of these two answers. And we see that our car reaches a maximum speed of 115 kilometers per hour when 𝑡 is equal to 15. Therefore, we’ve shown if the position function of a moving car is given by six plus five 𝑡 minus four 𝑡 squared where 𝑡 is greater than or equal to zero and 𝑡 is less than or equal to 15, then the maximum speed of the car on this interval is 115.