1.07i Differentiate x^n: for rational n and sums

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation \(y = f(x)\), where $$f(x) = 0.5e^x - x^2.$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).

1. Find an equation of the tangent to \(C\) at \(A\). [4]

The \(x\)-coordinate of \(B\) is approximately \(2.15\). A more exact estimate is to be made of this coordinate using iterations \(x_{n+1} = \ln g(x_n)\).

1. Show that a possible form for \(g(x)\) is \(g(x) = 4x\). [3]
2. Using \(x_{n+1} = \ln 4x_n\), with \(x_0 = 2.15\), calculate \(x_1\), \(x_2\) and \(x_3\). Give the value of \(x_3\) to 4 decimal places. [2]

$$f(x) = x^3 + x - 3.$$ The equation \(f(x) = 0\) has a root, \(\alpha\), between 1 and 2.

1. By considering \(f'(x)\), show that \(\alpha\) is the only real root of the equation \(f(x) = 0\). [3]
2. Taking 1.2 as your first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(f(x)\) to obtain a second approximation to \(\alpha\). Give your answer to 3 significant figures. [2]
3. Prove that your answer to part (b) gives the value of \(\alpha\) correct to 3 significant figures. [2]

Given that \(y = \tan x\),

1. find \(\frac{dy}{dx}\), \(\frac{d^2 y}{dx^2}\) and \(\frac{d^3 y}{dx^3}\). [3]
2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\) up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [3]
3. Hence show that \(\tan \frac{3\pi}{10} \approx 1 + \frac{\pi}{10} + \frac{\pi^2}{200} + \frac{\pi^3}{3000}\). [2]

\(y = 2x^2 - \frac{4}{\sqrt{x}} + 1\), \(x > 0\)

1. Find \(\frac{dy}{dx}\), giving each term in its simplest form. [3]
2. Find \(\frac{d^2y}{dx^2}\), giving each term in its simplest form. [2]

\(y = 7 + 10x^{\frac{1}{3}}\).

1. Find \(\frac{dy}{dx}\). [2]
2. Find \(\int y \, dx\). [3]

A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).

1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]

The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.

1. Find the \(x\)-coordinate of \(Q\). [2]
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]

This tangent intersects the coordinate axes at the points \(R\) and \(S\).

1. Find the length of \(RS\), giving your answer as a surd. [4]

Find \(\frac{dy}{dx}\) in each of the following cases:

1. \(y = \frac{(3x)^2 \times x^4}{x}\), [3]
2. \(y = ^3\sqrt{x}\), [3]
3. \(y = \frac{1}{2x^3}\). [2]

Find the coordinates of the points on the curve \(y = \frac{1}{3}x^3 + \frac{9}{x}\) at which the tangent is parallel to the line \(y = 8x + 3\). [10]

The points \(A(1, 3)\) and \(B(4, 21)\) lie on the curve \(y = x^2 + x + 1\).

1. Find the gradient of the line \(AB\). [2]
2. Find the gradient of the curve \(y = x^2 + x + 1\) at the point where \(x = 3\). [2]

A cuboid has a volume of \(8 \text{m}^3\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \text{m}^2\).

1. Show that \(A = 2x^2 + \frac{32}{x}\). [3]
2. Find \(\frac{dA}{dx}\). [3]
3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer. [4]

It is given that \(f(x) = \frac{6}{x^2} + 2x\).

1. Find \(f'(x)\). [3]
2. Find \(f''(x)\). [2]

Given that \(y = 6x^3 + \frac{4}{\sqrt{x}} + 5x\), find

1. \(\frac{\text{d}y}{\text{d}x}\), [4]
2. \(\frac{\text{d}^2y}{\text{d}x^2}\). [2]

A curve has equation \(y = 3x^3 - 7x + \frac{2}{x}\).

1. Verify that the curve has a stationary point when \(x = 1\). [5]
2. Determine the nature of this stationary point. [2]
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]

Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]

Differentiate with respect to \(x\) $$3x^2 - \sqrt{x} + \frac{1}{2x}.$$ [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 8x - x^{\frac{3}{2}}\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the \(x\)-coordinate of \(A\). [3]
2. Find the gradient of the tangent to the curve at \(A\). [4]

$$\text{f}(x) = \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
2. Show that $$\text{f}'(x) = \frac{(3x+4)(x-4)}{4x^{\frac{3}{2}}}.$$ [6]

Given that $$y = \frac{x^4 - 3}{2x^2},$$

1. find \(\frac{dy}{dx}\), [4]
2. show that \(\frac{d^2y}{dx^2} = \frac{x^4 - 9}{x^4}\). [2]

A curve has the equation \(y = \frac{x}{2} + 3 - \frac{1}{x}\), \(x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2.

1. Find the gradient of the curve at \(A\). [4]
2. Show that the tangent to the curve at \(A\) has equation $$3x - 4y + 8 = 0.$$ [3]

The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).

1. Find the coordinates of \(B\). [3]

Given that $$y = \sqrt{x} - \frac{4}{\sqrt{x}},$$

1. find \(\frac{dy}{dx}\). [3]
2. find \(\frac{d^2y}{dx^2}\). [2]
3. show that $$4x\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - y = 0.$$ [3]

Find \(\frac{dy}{dx}\) when

1. \(y = x - 2x^2\), [2]
2. \(y = \frac{3}{x^2}\). [2]

The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the \(x\)-coordinate of \(A\). [3]

The point \(B\) on the curve has \(x\)-coordinate 2.

1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]

Differentiate with respect to \(x\) $$\frac{6x^2 - 1}{2\sqrt{x}}.$$ [5]