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Find the equation of the curve for which y" = x, and which passes through the point (1, 2) with the slope of 5/2.
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
In order to get the value of arbitrary constant, substitute the value of the given point which is (1, 2) and the value of the given slope which is dy/dx = 5/2 to the above equation, we have
Hence, the above equation becomes
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 constant, substitute the value of the given point which is (1, 2) to the above equation, we have
Therefore, the equation of a curve is
