Question

find the equation of the curve that passes through (3,4) if its slope is given by the following equation.
\\( \frac { d y } { d x } = 2 x - 7 \\)
\\( y ( x ) = \square \\)

Explanation:

Step1: Integrate the derivative

Integrate \(\frac{dy}{dx}=2x - 7\) with respect to \(x\).
Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(y=\int(2x - 7)dx\).

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Step2: Find the value of \(C\)

Since the curve passes through the point \((3,4)\), substitute \(x = 3\) and \(y=4\) into \(y=x^{2}-7x + C\).

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Answer:

\(y(x)=x^{2}-7x + 16\)