A point moves along the curve y=(x-2)^2 such that the x coordinate is increasing at 2 units per second. At the moment x=1, how fast is its y coordinate changing.
y = (x-2)^2
y = x^2-4x -4
dy = 2xdx - 4dx
so,
dy/dx = 2x-4
dx/dt = 2
so,
dy/dt = dy/dx dx/dt
= 2(2x-4)
= 4x-8
for x=1 then
dy/dt = 4(1)-8 = -4 units per second
Well, well, well, we have a point on the move, huh? And not just any point, a point that loves curves! Let's find out how fast its y-coordinate is changing, shall we?
First, let's find the derivative of y = (x - 2)^2 with respect to x. Using the power rule, we have:
dy/dx = 2(x - 2)
Now, we know that at the moment x = 1, the x-coordinate is increasing at a rate of 2 units per second. This means dx/dt = 2.
To find dy/dt, we can use the chain rule:
dy/dt = dy/dx * dx/dt
Substituting the known values, we have:
dy/dt = 2(x - 2) * 2
Since we're interested in the rate when x = 1, let's substitute x = 1:
dy/dt = 2(1 - 2) * 2
dy/dt = -2 * 2
dy/dt = -4
Oh dear, it seems like the y-coordinate is changing at a rate of -4 units per second. That's a negative change, meaning it's decreasing. So, our point is curving and going down, but hey, at least it's moving!
Hope that puts a smile on your face!
To find how fast the y-coordinate is changing, we need to find dy/dt, which represents the rate of change of y with respect to time.
We are given the equation y = (x-2)^2 and are given that dx/dt (the rate of change of x with respect to time) is 2 units per second.
First, let's find dy/dx using implicit differentiation:
dy/dx = 2(x-2)
Now, we can find dy/dt using the chain rule:
dy/dt = dy/dx * dx/dt
Substituting the given values into the formula, we have:
dy/dt = 2(1-2) * 2
Simplifying, we get:
dy/dt = -2 * 2
dy/dt = -4
Therefore, the y-coordinate is changing at a rate of -4 units per second when x=1.
To find how fast the y-coordinate is changing, we need to find dy/dt, which represents the rate of change of y with respect to time (t).
Given the equation y = (x-2)^2, we can differentiate both sides with respect to x to find dy/dx:
dy/dx = d/dx((x-2)^2)
To differentiate (x-2)^2, we can use the chain rule. Let's denote u as the function (x-2) and apply the chain rule:
dy/dx = d/du(u^2) * du/dx
Now let's differentiate u^2 with respect to u:
dy/dx = 2u * du/dx
But u is equal to (x-2), so we substitute it back in:
dy/dx = 2(x-2) * du/dx
Now we need to rearrange the equation and solve for du/dx:
dy/dx = 2(x-2) * du/dx
du/dx = dy/dx / (2(x-2))
We need to find the value of du/dx when x = 1. Since dx/dt = 2 units per second, du/dt = du/dx * dx/dt:
du/dt = (dy/dx / (2(x-2))) * dx/dt
Substituting x = 1 and dx/dt = 2:
du/dt = (dy/dx / (2(1-2))) * 2
Simplifying:
du/dt = -2dy/dx
Finally, we plug in the value of dy/dx. Since dy/dx = 2(x-2), when x = 1:
dy/dx = 2(1-2) = -2
Plugging this into the equation:
du/dt = -2(-2) = 4 units per second
Therefore, the y-coordinate is changing at a rate of 4 units per second when x = 1.