Suppose the position of an object moving horizontally along a ...

Question

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

On what intervals is the speed increasing?

f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A car travels along a straight road. And its position at time T seconds is given by the function PF T is equal to 4 T 2 minus 12 T plus 9, where P is in kilometers and 0 is less than or equal to T and T is less than or equal to 3. On which intervals is the car's speed increasing. So that's our end goal. Our angle, the final answer we're ultimately trying to solve for is we're trying to figure out on which intervals this particular car's speed is increasing. So we're trying to figure out the intervals at which the car speed is increasing. So now that we know what we're solving for, let's take a moment to read off our multiple choice sensors to see what our final answer might be, noting that a parenthesis denotes open and a square bracket denotes closed. So A is parenthesis 1.5.3 bracket. B is bracket 0.3 bracket. C is parenthesis 23 and D is 0.1.5 bracket. Awesome. So first off, we need to define our position function, which is PFT and PFT is given to us by the problem itself as 4T2 minus 12T plus 9. So now our next step is we need to find the speed, and in order to find the speed, we need to find the first derivative of the position function, which will represent our velocity function VFT. So now we can calculate VFT and as we should note and recall, VAT is equal to DP of T by DT, which is equal to 8T minus 12. Fantastic. So now we need to determine when the speed is increasing, and in order to do that, as we should recall, we need to take the second derivative of the position function. Which represents the acceleration AFT. So AFT, the acceleration is equal to D V T DV of T, so DV. Of T by DT and this is equal to 8. Fantastic. So now we need to note and recall that since A of T is equal to 8, is positive for all values of T in the given interval, we also need to satisfy the condition that the velocity function should also be positive, meaning that the speed is increasing as long as both the acceleration and velocity have the same direction. So therefore we can solve that V of T is greater than 0. So substituting in the given velocity function from our second step, which our second step was when we solved for a V of T. And our first step is when we determined the position function, and our third step is when we found the acceleration. So once again, when we substitute the given velocity function from step 2, we end up with the following inequality that 8, so 8T minus 12 is greater than 0, fantastic. So now our next step is we need to rearrange this equation to isolate and solve for T. So when we start to do that, we will find that 8T. is greater than 12 and getting T all by itself, you will find that T is greater than 12 divided by 8, or this will mean that T is, once again, T is greater than To 1.5. Fantastic. So combining the given interval, so combining this with our given interval, we need to make sure that the time doesn't exceed 3 seconds, which will mean that we could say that T belongs to parenthesis 1.5.3 square bracket, which will be our final answer. Hooray, we did it. And this corresponds with multiple choice answer letter A, which once again is parenthesis 1.5.3 square bracket. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

Key Concepts

Position Function

Velocity and Speed

Increasing Functions

Watch next

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.On what intervals is the speed increasing?f(t) = 2t2 - 9t + 12; 0 ≤ t ≤ 3

Key Concepts

Position Function

Velocity and Speed

Increasing Functions

Watch next

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3