A particle moves along the curve y = sqr(1+x^3). As it reaches the point (2,3), the y-coordinate is increasing at a rate of 4cm/s. How fast is the x-coordinate of the point changing at that instant?
To find the rate at which the x-coordinate is changing, we need to use the chain rule of differentiation. The chain rule states that if we have a composite function y = f(g(x)), then dy/dx = dy/dg * dg/dx.
Here, we have y = sqrt(1 + x^3), so we can consider y as the composite function of two functions: y = f(g(x)), where g(x) = 1 + x^3 and f(u) = sqrt(u).
First, let's find dy/dg, which is the derivative of f(u) = sqrt(u) with respect to u. We can use the power rule for differentiation:
df(u)/du = (1/2) * u^(-1/2)
Now, let's find dg/dx, which is the derivative of g(x) = 1 + x^3 with respect to x. Again, we can use the power rule:
dg(x)/dx = 3x^2
Now, we can apply the chain rule and multiply these two derivatives together to find dy/dx:
dy/dx = (dy/dg) * (dg/dx) = [(1/2) * (1 + x^3)^(-1/2)] * (3x^2)
Now, we can substitute the given values into this expression. At the point (2,3), the y-coordinate is increasing at a rate of 4 cm/s, so dy/dt = 4 cm/s. We want to find dx/dt, which represents the rate at which the x-coordinate is changing. Since dy/dx = dy/dt / dx/dt, we can rearrange the equation to solve for dx/dt:
(dy/dx) * (dx/dt) = dy/dt
dx/dt = dy/dt / dy/dx
Substituting the known values:
dx/dt = 4 cm/s / [(1/2) * (1 + 2^3)^(-1/2)] * (3 * 2^2)
Simplifying:
dx/dt = 4 cm/s / [(1/2) * (1 + 8)^(-1/2)] * 12
Calculating further:
dx/dt = 4 cm/s / [(1/2) * 9^(-1/2)] * 12
= 4 cm/s / 3/2 * 12
= 8 cm/s
Therefore, the x-coordinate of the point is changing at a rate of 8 cm/s at the instant when the y-coordinate is increasing at 4 cm/s.
To find the rate at which the x-coordinate is changing, we can use the chain rule.
Given that the particle moves along the curve y = √(1+x^3), we can differentiate both sides of the equation with respect to time.
dy/dt = d/dt (√(1+x^3))
Since we are given that dy/dt = 4 cm/s, we can substitute this value into the equation:
4 = d/dt (√(1+x^3))
Now, let's find the derivative of the right side of the equation using the chain rule.
To differentiate √(1+x^3), we can let u = 1+x^3 and apply the power rule.
du/dx = 3x^2
Now, we can rewrite the equation as:
4 = d/du (√u) * du/dx
To find d/du (√u), we will differentiate √u with respect to u:
d/du (√u) = 1/2√u
Substituting this back into the equation:
4 = 1/2√u * du/dx
We know that u = 1+x^3, so:
4 = 1/2√(1+x^3) * du/dx
Now we can solve for du/dx:
du/dx = 8/√(1+x^3)
To find how fast the x-coordinate is changing at the point (2,3), we need to evaluate du/dx when x = 2:
du/dx = 8/√(1+2^3)
= 8/√(1+8)
= 8/√9
= 8/3
Therefore, the x-coordinate of the point is changing at a rate of 8/3 cm/s at that instant.