OCR MEI C2 (Core Mathematics 2)

Differentiate \(x + \sqrt{x^3}\). [4]

The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]

A and B are points on the curve \(y = 4\sqrt{x}\). Point A has coordinates \((9, 12)\) and point B has \(x\)-coordinate \(9.5\). Find the gradient of the chord AB. The gradient of AB is an approximation to the gradient of the curve at A. State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation. [3]

Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]

Find the set of values of \(x\) for which \(x^2 - 7x\) is a decreasing function. [3]

Differentiate \(10x^4 + 12\). [2]

\includegraphics{figure_7} Fig. 10 shows a solid cuboid with square base of side \(x\) cm and height \(h\) cm. Its volume is \(120\) cm\(^3\).

1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A\) cm\(^2\), of the cuboid is given by $$A = 2x^2 + \frac{480}{x}.$$ [3]
2. Find \(\frac{dA}{dx}\) and \(\frac{d^2A}{dx^2}\). [4]
3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case. [5]

Differentiate \(6x^{\frac{5}{2}} + 4\). [2]

A is the point \((2, 1)\) on the curve \(y = \frac{4}{x^2}\). B is the point on the same curve with \(x\)-coordinate \(2.1\).

1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places. [2]
2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A. [1]
3. Use calculus to find the gradient of the curve at A. [2]

The gradient of a curve is given by \(\frac{dy}{dx} = x^2 - 6x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\). [3]

A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]