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An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If $ t $ represents the time in minutes since the plane has left the terminal building, let $ x(t) $ be the horizontal distance traveled and $ y(t) $ be the altitude of the plane.
(a) Sketch a possible graph of $ x(t) $.(b) Sketch a possible graph of $ y(t) $.(c) Sketch a possible graph of the ground speed.(d) Sketch a possible graph of the vertical velocity.
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Calculus 1 / AB
Calculus 2 / BC
Functions and Models
Four Ways to Represent a Function
Functions of Several Variables
Oregon State University
A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.
In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.
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Okay, so we're going to solve the fallen problem. An air plane takes off from an airport in lands. An hour later, another airport four hundred miles away. If t represents the time in minutes since the plane has left the terminal building, let X of TB, the horizontal distance traveled and wives he be the altitude of the plane. So our first task is to sketch the possible graph of exit E. So recall, except he's the horizontal distance traveled. Okay, So to do this, we're going to start by, uh, see in a drawer X and y cornets. Well, actually, in this case, it would be our t for time coordinate, and our ex would be as a function of tea. Okay, so X of tea again is going to be our horizontal distance. While our total distance we're going to travel is four hundred miles. And we're told that we're gonna be covering four hundred miles in one hour. So at the end of sixty minutes, our trip will be over. Eso an example of what this graft might look like would be. We started at zero from the origin, So zero miles covered zero minutes traveled we're going to take off, and then our velocity will increase and then eventually stabilize as the plane starts cruising that eventually it will start decreasing. Yet until we land. And at this point, we will have reached our for a hundred miles in sixty minutes. Okay? Task be is to sketch the possible graph of variety. Recall that Why? Lt is the altitude of the plane. Okay, so our plane is again. So now this is t on DH. This is why, Yeah, Every goes. Why? Okay, so, again, this is going to take a total of sixty minutes, and we're going to start off from the ground, and eventually we're going to really reach cruising altitude at which top points thing else you will not increase. Eventually, we're going to start the panther departure, and we're going to land after sixty minutes. So that is an example of the graph for what lives he should look like. Okay, now we're going to sketch a possible graph of the ground speed. So this is the speed claimed, respect you the ground. And this is going to be similar toe horizontal velocity. So we're gonna have a graph as a function of tea. And now this is our ground speed. So we don't have variable for this has yet, but it's going to, uh, it's going to look something like this. We're going to increase stabilizers. We reach cruising altitude and then decrease. Hey, finally, our last passes a sketch, a possible graph of the vertical velocity. Okay, so now we are looking at a graph, and here we're going to have except tea. So recall again, A Except tea is the horizontal distance traveled. And why a t he else to the plane? And now their velocity increases. Ah, that plane's moving forward. And they were going to reach our cruising altitude, and then we're going to decrease for landing here in the middle is going to be a cruising altitude. Okay, so these are for grass that represents different situations that we've been presented with
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