The slope of the tangent line to a curve is given by f'(x)=4x^2 + 7x -9. If the point ​(0,6​) is on the​ curve, find an equation of the curve.

y-y1=m(x-x1)

if

f'(x)=4x^2 + 7x -9
then
f(x) = y = (4/3) x^3 + (7/2)x^2 - 9 x + constant
use point to get constant
6 = 0 + constant

To find the equation of the curve, we need to integrate the given derivative function. The integral of f'(x) will give us the original function f(x), which represents the curve.

The given derivative function is f'(x) = 4x^2 + 7x - 9.

To integrate f'(x), we apply the power rule of integration. The power rule states that integrating x^n gives us (1/(n+1)) * x^(n+1), where n is any real number except -1.

Using the power rule, we integrate 4x^2 as follows:
∫ 4x^2 dx = (4/3) * x^3 + C1,

where C1 is the constant of integration.

Next, we integrate 7x as follows:
∫ 7x dx = (7/2) * x^2 + C2,

where C2 is another constant of integration.

Finally, integrating -9 gives us:
∫ -9 dx = -9x + C3,

where C3 is the third constant of integration.

Combining these results, we have:
f(x) = (4/3) * x^3 + (7/2) * x^2 - 9x + C,

where C = C1 + C2 + C3.

Since the point (0, 6) is on the curve, we can use it to find the value of C:
f(0) = (4/3) * 0^3 + (7/2) * 0^2 - 9 * 0 + C = 6.

Simplifying, we get:
C = 6.

Therefore, the equation of the curve is given by:
f(x) = (4/3) * x^3 + (7/2) * x^2 - 9x + 6.

To find an equation of the curve, we need to integrate the derivative of the curve, which is given as f'(x) = 4x^2 + 7x - 9.

1. Integrate f'(x) to find the function f(x):
To integrate f'(x), we need to find the antiderivative of 4x^2 + 7x - 9, which will give us f(x).

∫(4x^2 + 7x - 9) dx = F(x) + C

Where F(x) represents the antiderivative of the function, and C is the constant of integration.

Using the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1), we can find the antiderivative:

F(x) = (4/3)x^3 + (7/2)x^2 - 9x + C

Here, C represents the constant of integration.

2. Find the constant of integration:
We are given that the point (0,6) lies on the curve. So, we can use this information to find the constant of integration C.

Substitute x = 0 and y = 6 into the equation obtained from integrating:
6 = (4/3)(0)^3 + (7/2)(0)^2 - 9(0) + C
6 = C

Hence, the constant of integration is C = 6.

3. Write the equation of the curve:
Now that we have the constant of integration, we can write the equation of the curve:

f(x) = (4/3)x^3 + (7/2)x^2 - 9x + 6

This is the equation of the curve.