The function \(f\) is defined by \(f(x) = \frac{2}{(x+2)^2}\) for \(x > -2\).
(a) Find \(\int_{1}^{\infty} f(x) \, dx\).
(b) The equation of a curve is such that \(\frac{dy}{dx} = f(x)\). It is given that the point \((-1, -1)\) lies on the curve.
Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}\). The curve passes through the point \(P(2, 8)\).
(a) Find the equation of the normal to the curve at \(P\).
(b) Find the equation of the curve.
The point (4, 7) lies on the curve \(y = f(x)\) and it is given that \(f'(x) = 6x^{-\frac{1}{2}} - 4x^{-\frac{3}{2}}\).
Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{(x-3)^2} + x\). It is given that the curve passes through the point (2, 7).
Find the equation of the curve.
The equation of a curve is such that \(\frac{dy}{dx} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}\). It is given that the point (4, 7) lies on the curve.
Find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = x^3 - \frac{4}{x^2}\). The point \(P(2, 9)\) lies on the curve.
Find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = \frac{12}{(2x+1)^2}\). The point (1, 1) lies on the curve. Find the coordinates of the point at which the curve intersects the x-axis.
A curve passes through the point (4, -6) and has an equation for which \(\frac{dy}{dx} = x^{-\frac{1}{2}} - 3\). Find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = -x^2 + 5x - 4\).
Given that the curve passes through the point (6, 2), find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = \frac{8}{\sqrt{4x + 1}}\). The point \((2, 5)\) lies on the curve. Find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = \frac{8}{(5 - 2x)^2}\). Given that the curve passes through (2, 7), find the equation of the curve.
A curve for which \(\frac{dy}{dx} = 3x^2 - \frac{2}{x^3}\) passes through \((-1, 3)\). Find the equation of the curve.
A curve has a stationary point at \(2, -10\) and is such that \(\frac{d^2y}{dx^2} = 6x\).
(a) Find \(\frac{dy}{dx}\).
(b) Find the equation of the curve.
(c) Find the coordinates of the other stationary point and determine its nature.
(d) Find the equation of the tangent to the curve at the point where the curve crosses the y-axis.
The function \(f\) is such that \(f'(x) = 3x^2 - 7\) and \(f(3) = 5\). Find \(f(x)\).
A curve is such that \(\frac{dy}{dx} = (2x + 1)^{\frac{1}{2}}\) and the point \((4, 7)\) lies on the curve. Find the equation of the curve.
The function \(f\) is such that \(f'(x) = 5 - 2x^2\) and \((3, 5)\) is a point on the curve \(y = f(x)\). Find \(f(x)\).
A curve has equation \(y = f(x)\). It is given that \(f'(x) = x^{-\frac{3}{2}} + 1\) and that \(f(4) = 5\). Find \(f(x)\).
A curve has equation \(y = f(x)\). It is given that \(f'(x) = \frac{1}{\sqrt{x+6}} + \frac{6}{x^2}\) and that \(f(3) = 1\). Find \(f(x)\). [5]
A curve is such that \(\frac{dy}{dx} = \sqrt{2x + 5}\) and \((2, 5)\) is a point on the curve. Find the equation of the curve.
A curve is such that \(\frac{dy}{dx} = \frac{6}{x^2}\) and \((2, 9)\) is a point on the curve. Find the equation of the curve.