A curve \(y=\mathrm{f}(x)\) is such that \(\mathrm{f}''(x)=(2x+5)^{-\frac32}\). The curve has gradient \(\frac23\) at the point \((2,2)\).

(a) Find the coordinates of the stationary point on the curve.

(b) Determine the nature of this stationary point.

Given that

\(f''(x)=(5x+2)^{-2/5},\qquad f'(6)=\frac{17}{3}\)

and

\(f(6)=\frac{26}{3},\)

find an expression for \(f(x)\).

Given that

\(f''(x)=6(3x+4)^{-\frac12},\)

\(f'(4)=18\)

and

\(f(4)=\frac{512}{9},\)

find \(f(x)\).

A curve with equation \(y=\mathrm f(x)\) is such that

\(\frac{d^2y}{dx^2}=(2x+3)^{-\frac12}+5\)

for \(x\gt 0\). The curve has gradient \(10\) at the point \(\left(3,\frac{19}{2}\right)\).

(a) Show that, when \(x=11\), \(\frac{dy}{dx}=52\).

(b) Find \(\mathrm f(x)\).

It is given that

\(\frac{d^2y}{dx^2}=e^{2x}+\frac1{(x+1)^2}\)

for \(x\gt -1\).

(a) Find an expression for \(\frac{dy}{dx}\), given that \(\frac{dy}{dx}=2\) when \(x=0\).

(b) Find an expression for \(y\), given that \(y=4\) when \(x=0\).

A curve is such that

\(\frac{d^2y}{dx^2}=5\cos2x.\)

This curve has a gradient of \(\frac34\) at the point \(\left(-\frac{\pi}{12},\frac{5\pi}{4}\right)\). Find the equation of this curve.

A curve is such that

\(\frac{d^2y}{dx^2}=2\sin\left(x+\frac{\pi}{3}\right).\)

Given that the curve has a gradient of \(5\) at the point \(\left(\frac{\pi}{3},\frac{5\pi}{3}\right)\), find the equation of the curve.

A curve is such that

\(\frac{d^2y}{dx^2}=2(3x-1)^{-\frac23}.\)

Given that the curve has a gradient of \(6\) at the point \((3,11)\), find the equation of the curve.

A curve is such that

\(\frac{d^2y}{dx^2}=(2x-5)^{-1/2}.\)

Given that the curve has a gradient of \(6\) at the point \(\left(\dfrac92,\dfrac23\right)\), find the equation of the curve.

A curve is such that

\(\frac{d^2y}{dx^2}=(2x-5)^{-1/2}.\)

Given that the curve has a gradient of \(6\) at the point \(\left(\dfrac92,\dfrac23\right)\), find the equation of the curve.

Find the equation of the curve which has a gradient of \(4\) at the point \((0,-3)\) and is such that

\(\frac{d^2y}{dx^2}=5+e^{2x}.\)

\(\frac{d^2y}{dx^2}=2x+\frac{3}{(x+1)^4}.\)

(i) Find \(\dfrac{dy}{dx}\), given that \(\dfrac{dy}{dx}=1\) when \(x=1\).

(ii) Find \(y\) in terms of \(x\), given that \(y=3\) when \(x=1\).

Find the equation of the curve which passes through the point \((2,17)\) and for which

\(\frac{dy}{dx}=4x^3+1.\)

A curve is such that \(\dfrac{dy}{dx}=10e^{5x}+3\). It is given that the curve passes through the point \((0,9)\). Find the equation of the curve.

Find \(y\) in terms of \(x\), given that \(\dfrac{d^2y}{dx^2}=6x+\dfrac2{x^3}\) and that when \(x=1\), \(y=3\) and \(\dfrac{dy}{dx}=1\).

The gradient of the normal to a curve at the point with coordinates \((x,y)\) is given by \(\dfrac{\sqrt x}{1-3x}\).

(i) Find the equation of the curve, given that the curve passes through the point \((1,-10)\).

(ii) Find, in the form \(y=mx+c\), the equation of the tangent to the curve at the point where \(x=4\).