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Find the equation of the curve for which y" = 6x², and which passes through the points (0, 2) and (-1, 3).
Solution:
The concavity of a curve is equal to the second derivative of a curve with respect to x. In this case, y" = d²y/ dx². Let's consider the given concavity of a curve
We can rewrite the above equation as follows
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
Since the slope of a curve is not given in the problem but the two points are given, then we have to continue the integration until we get an equation in terms of x and y. Let's consider the equation above
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
In order to get the value of arbitrary constants, we need to use the coordinates of two points so that we can form the two equations, two unknowns.
By using the point (0, 2), substitute the value of x and y to the above equation, we have
By using the point (-1, 3), substitute the value of x and y to the above equation, we have
but
then the above equation becomes
Therefore, the equation of a curve is
