To find the equation of a curve that passes through the point [tex]\((-2, 5)\)[/tex] and has a gradient described by the function [tex]\(6x^2 + 8x - 3\)[/tex], follow these steps:
1. Find the antiderivative (integral) of the gradient function:
The gradient function [tex]\(6x^2 + 8x - 3\)[/tex] is given as the derivative of our unknown curve function. To find the curve, we need to integrate the gradient function.
[tex]\[
\int (6x^2 + 8x - 3) \, dx
\][/tex]
2. Compute the indefinite integral:
Integrate each term separately:
[tex]\[
\int 6x^2 \, dx = 2x^3
\][/tex]
[tex]\[
\int 8x \, dx = 4x^2
\][/tex]
[tex]\[
\int -3 \, dx = -3x
\][/tex]
Combining these, we find the general form of the curve to be:
[tex]\[
2x^3 + 4x^2 - 3x + C
\][/tex]
Here, [tex]\(C\)[/tex] is the constant of integration that we need to determine.
3. Determine the constant of integration [tex]\(C\)[/tex]:
We use the given point [tex]\((-2, 5)\)[/tex] to find the value of [tex]\(C\)[/tex]. Substituting [tex]\(x = -2\)[/tex] and [tex]\(y = 5\)[/tex] into the equation of the curve:
[tex]\[
5 = 2(-2)^3 + 4(-2)^2 - 3(-2) + C
\][/tex]
Simplify the equation:
[tex]\[
5 = 2(-8) + 4(4) - 3(-2) + C
\][/tex]
[tex]\[
5 = -16 + 16 + 6 + C
\][/tex]
[tex]\[
5 = 6 + C
\][/tex]
Solving for [tex]\(C\)[/tex]:
[tex]\[
C = 5 - 6 = -1
\][/tex]
4. Write the final equation of the curve:
Substitute [tex]\(C = -1\)[/tex] back into the general form of the curve:
[tex]\[
y = 2x^3 + 4x^2 - 3x - 1
\][/tex]
Thus, the equation of the curve that passes through the point [tex]\((-2, 5)\)[/tex] and has a gradient of [tex]\(6x^2 + 8x - 3\)[/tex] at any point on the curve is:
[tex]\[
y = 2x^3 + 4x^2 - 3x - 1
\][/tex]
