The equation of a curve is such that \( \frac{dy}{dx}=2x-6x^{\frac12} \). The curve passes through the point \( (4,-9) \).
Find the equation of the curve.
Solution
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Integrate \( \frac{dy}{dx} \):
\( y=\int \left(2x-6x^{\frac12}\right)\,dx \).
\( y=x^2-4x^{\frac32}+C \).
Use the point \( (4,-9) \):
\( -9=4^2-4(4^{\frac32})+C \).
\( -9=16-4(8)+C=-16+C \).
So \( C=7 \).
Answer: \( y=x^2-4x^{\frac32}+7 \).
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