A particle moves along the curve y = 3x2 + 1 in such a way that the y value is increasing at the rate of 3 units per second. At what rate is x changing when x = 2?
y = 3x^2 + 1
dy/dt = 6x dx/dt
so,
3 = 6*2 dx/dt
dx/dt = 1/4
To find the rate at which x is changing when x = 2, we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to time (t) using the chain rule.
d/dt(y) = d/dt(3x^2) + d/dt(1)
Step 2: Since dy/dt is given as 3 units per second, substitute the value.
3 = d/dt(3x^2) + 0
Step 3: Differentiate 3x^2 with respect to x using the power rule.
3 = 6x * (dx/dt)
Step 4: Substitute x = 2 to find the value of dx/dt.
3 = 6(2) * (dx/dt)
Step 5: Solve for dx/dt.
dx/dt = 3/(6*2) = 1/4
Therefore, when x = 2, the rate at which x is changing is 1/4 units per second.
To find the rate at which x is changing, we need to differentiate the equation of the curve with respect to time. Let's start by differentiating the equation y = 3x^2 + 1 with respect to time (t).
dy/dt = d(3x^2 + 1)/dt
Now, let's differentiate each term separately. The derivative of 3x^2 with respect to t is 6x(dx/dt), and the derivative of 1 with respect to t is 0.
Therefore:
dy/dt = 6x(dx/dt)
Now, we are given that dy/dt (rate of change of y) is 3 units per second. So we can substitute this value into the equation:
3 = 6x(dx/dt)
Now we need to find dx/dt (rate of change of x) when x = 2. We can substitute x = 2 into the equation:
3 = 6 * 2(dx/dt)
3 = 12(dx/dt)
To find dx/dt, we can divide both sides of the equation by 12:
3/12 = dx/dt
dx/dt = 1/4
So, when x = 2, the rate of change of x is 1/4 units per second.