1. [-/1 Points] DETAILS MY NOTES A drone moves on a straight...

1. [-/1 Points] DETAILS MY NOTES A drone moves on a straight line. We know that its position is modeled by $\frac{dx}{dt} = t + 7x$, $x(0) = 4$ y(x) = Give the largest interval $I$ over which the solution is defined. (Enter your answer using interval notation.) $I = $

Transcript

00:01 Okay, so we want to figure out where is this object moving right from its displacement function, x of t.

00:07 And so the way that we're going to do this is we're going to find our velocity equation.

00:11 So v of t here is equal to the derivative of x of t, x prime of t, and then we're going to set it equal to zero and find where t is equal to zero.

00:20 And then we'll go from there.

00:21 I'll explain it from there.

00:22 So the derivative function, x prime of t, is going to be equal to the derivative of t to the fourth, which is 14.

00:30 T cubed minus 12 times the derivative of t cubed which be 12 times 3 t squared or 36 t squared and then plus 28 28 times the derivative of t squared which would be 56 t and so now that we have our velocity equation let's go ahead and set it equal to zero and i'm also going to factor out a 4 t on this right side so we're going to have 4 t times t squared minus 9t and then plus 14.

01:08 And so t squared minus 9t plus 14, that actually factors as well into t minus 2 times t minus 7 is equal to 0.

01:19 So that means at t is equal to zero, at t is equal to 2 and at t is equal to seven are the values where our derivative or our velocity equation is equal to zero.

01:28 And the reason that we want to have these values is because for something to go from positive to negative, it needs to pass the x -axis.

01:38 It needs to have a value where it's going to be equal to zero as long as it's a continuous equation.

01:43 In this case, our equation or our velocity equation is continuous.

01:47 So that means that in order for us to go from positive to negative, we have to cross the x -axis.

01:52 So really, we just need to look at values of t that are less than zero, that are between zero and two, and then we actually shouldn't do that.

02:05 And then for values that are between 2 and 7, and then for values that are greater than 7.

02:12 And then whatever value or whatever sign our derivative is at these values, if it's positive for t less than 0, then we're going to know that we're going right there.

02:24 If it's positive in this one, then we know we're going right and so on.

02:27 So let's go ahead and just do this for values of t that are less than zero...

Question

1. [-/1 Points] DETAILS MY NOTES A drone moves on a straight line. We know that its position is modeled by $\frac{dx}{dt} = t + 7x$, $x(0) = 4$ y(x) = Give the largest interval $I$ over which the solution is defined. (Enter your answer using interval notation.) $I = $

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