A curve with equation $y = f(x)$ is such that \( \frac{dy}{dx} = -\frac{3}{x^2} \), where $x > 0$ - Scottish Highers Maths - Question 6 - 2022
Question 6
A curve with equation $y = f(x)$ is such that \( \frac{dy}{dx} = -\frac{3}{x^2} \), where $x > 0$.
The curve passes through the point (3, 6).
Express $y$ in terms ... show full transcript
Worked Solution & Example Answer:A curve with equation $y = f(x)$ is such that \( \frac{dy}{dx} = -\frac{3}{x^2} \), where $x > 0$ - Scottish Highers Maths - Question 6 - 2022
Step 1
Write in integrable form
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Answer
We start with the equation:
[ \frac{dy}{dx} = -\frac{3}{x^2} ]
This can be rewritten in integrable form as:
[ dy = -\frac{3}{x^2} dx ]
Step 2
Integrate one term
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Answer
Now, we integrate both sides:
[ \int dy = \int -\frac{3}{x^2} dx ]
The left side integrates to y, and the right side integrates as follows:
[ \int -\frac{3}{x^2} dx = 3x^{-1} = -\frac{3}{x} + C ]
So we have:
[ y = -\frac{3}{x} + C ]
Step 3
Substitute for $y$ and $x$
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Answer
Using the information given that the curve passes through the point (3, 6), we can substitute:
[ 6 = -\frac{3}{3} + C ]
This simplifies to:
[ 6 = -1 + C ]
Thus,
[ C = 7 ]
Step 4
State expression for $y$
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Answer
Substituting C back into the equation for y, we obtain:
[ y = -\frac{3}{x} + 7 ]
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