The gradient of a curve is defined by

dy/dx = 3x^(1/2) - 6

Given the point (9, 2) lies on the curve, find the equation of the curve

y' = 3x^(1/2) - 6

y = 2x^(3/2) - 6x + C
y(9) = 2, so

2*27 - 6*9 + C = 2
C = 2

y = 2x^(3/2) - 6x + 2

Well, if we integrate both sides of the equation with respect to x, we can find the equation of the curve. Let's give it a go!

Integrating the left side, we get:
∫dy = ∫(3x^(1/2) - 6) dx

Integrating the right side, we have:
y = ∫3x^(1/2) dx - ∫6 dx

Now let's solve these integrals separately:

∫3x^(1/2) dx = 2x^(3/2) + C1

And ∫6 dx = 6x + C2

Combining them, we have:
y = 2x^(3/2) + 6x + C1 + C2

Since the point (9, 2) lies on the curve, we can substitute these values into the equation to find C1 + C2:

2 = 2(9)^(3/2) + 6(9) + C1 + C2

Solving this equation will yield the values of C1 and C2. Once we have those, we can plug them back into the equation to obtain the final equation of the curve.

To find the equation of the curve when the gradient is given, you need to integrate the gradient function with respect to x to get the original function.

Integrating both sides of the equation dy/dx = 3x^(1/2) - 6 with respect to x:

∫dy = ∫(3x^(1/2) - 6) dx

Integrating the right-hand side using the power rule of integration:
∫(3x^(1/2) - 6) dx = ∫3x^(1/2) dx - ∫6 dx

This simplifies to:
y = (2/3)x^(3/2) - 6x + C

Where C is the constant of integration.

To find the value of C, we can substitute the given point (9, 2) into the equation. So we have:
2 = (2/3)(9)^(3/2) - 6(9) + C

Simplifying this equation:

2 = (2/3)(27) - 54 + C
2 = 18 - 54 + C
2 = -36 + C

Now, solving for C:
C = 2 + 36
C = 38

Therefore, the equation of the curve is:
y = (2/3)x^(3/2) - 6x + 38

To find the equation of the curve, we need to integrate the expression for dy/dx with respect to x.

Integrating both sides of the equation, we get:

∫(dy/dx) dx = ∫(3x^(1/2) - 6) dx

This simplifies to:

∫dy = ∫(3x^(1/2) - 6) dx

Integration gives us:

y = 2x^(3/2) - 6x + C

To find the constant of integration (C), we can use the given point (9, 2) that lies on the curve.

Substituting x = 9 and y = 2 into the equation, we get:

2 = 2(9)^(3/2) - 6(9) + C

Simplifying further, we have:

2 = 2(27) - 54 + C

2 = 54 - 54 + C

C = 2

Therefore, the equation of the curve is:

y = 2x^(3/2) - 6x + 2