A particle is moving along the curve y = 4x^2 + 1 in such a way that the y coordinate is increasing at a rate of ½ units per second. At what rate is the x coordinate changing at the instant x = 2?
y = 4x^2 + 1
dy/dt = 8x dx/dt
when x = 2, dy/dt = 1/2
sub that in....
1/2 = 16 dx/dt
dx/dt = 1/32
Well, if the y-coordinate is increasing at a rate of ½ units per second, we can use the power rule of differentiation to find the rate at which the x-coordinate is changing. The derivative of y = 4x^2 + 1 is dy/dx = 8x.
Now, we know that dy/dt = 1/2, since the y-coordinate is increasing at a rate of ½ units per second. We want to find dx/dt, the rate at which the x-coordinate is changing.
To find dx/dt, we can use the chain rule: dx/dt = (dx/dy) * (dy/dt). Since dx/dy is the reciprocal of dy/dx, we have dx/dt = (1/8x) * (1/2) = 1/16x.
Substituting x = 2 into the equation, we have dx/dt = 1/16(2) = 1/32 units per second.
So, the x-coordinate is changing at a rate of 1/32 units per second when x = 2. Keep on rollin', Mr. Particle!
To find the rate at which the x-coordinate is changing, we can use implicit differentiation.
Given the equation y = 4x^2 + 1, we need to find the derivative, dy/dx, with respect to time (t).
Differentiating both sides of the equation with respect to t using the chain rule, we get:
dy/dt = d/dt (4x^2 + 1)
Now, let's differentiate each term separately:
Using the power rule, d/dt (4x^2) = 8x (dx/dt)
Differentiating the constant term, d/dt (1) = 0
Therefore, we have:
dy/dt = 8x (dx/dt)
Now, we are given that dy/dt (the rate at which the y-coordinate is changing) is increasing at a rate of 1/2 units per second. So dy/dt = 1/2.
Substituting into the equation, we get:
1/2 = 8x (dx/dt)
Since we are interested in finding the rate at which the x-coordinate is changing (dx/dt) at the instant x = 2, we can substitute x = 2 into the equation:
1/2 = 8(2) (dx/dt)
Simplifying further:
1/2 = 16 (dx/dt)
Now, we can solve for dx/dt:
dx/dt = (1/2) / 16
Simplifying:
dx/dt = 1/32
Therefore, the rate at which the x-coordinate is changing at the instant x = 2 is 1/32 units per second.