1.07q Product and quotient rules: differentiation

The function f(x) is defined as \(f(x) = \frac{\ln x}{x}\). The graph of the function is shown in Fig. 6. \includegraphics{figure_6}

1. Give the coordinates of the point, P, where the curve crosses the \(x\)-axis. [1]
2. Use calculus to find the coordinates of the stationary point, Q, and show that it is a maximum. [6]

The functions f(x) and g(x) are defined by $$f(x) = x^2, \quad g(x) = 2x - 1,$$ for all real values of \(x\).

1. State the ranges of f(x) and g(x). Explain why f(x) has no inverse. [3]
2. Find an expression for the inverse function g\(^{-1}\)(x) in terms of \(x\). Sketch the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\) on the same axes. [4]
3. Find expressions for gf(x) and fg(x). [2]
4. Solve the equation gf(x) = fg(x). Sketch the graphs of \(y = gf(x)\) and \(y = fg(x)\) on the same axes to illustrate your answer. [4]
5. Show that the equation f(x + a) = g\(^{-1}\)(x) has no solution if \(a > \frac{1}{4}\). [5]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(x^2 e^{-x}\) [3]
3. \(\frac{\sin x}{3 + 2\cos x}\) [4]

$$f(x) = 2x^2 + 3 \ln (2 - x), \quad x \in \mathbb{R}, \quad x < 2.$$

1. Show that the equation \(f(x) = 0\) can be written in the form $$x = 2 - e^{kx^2},$$ where \(k\) is a constant to be found. [3]

The root, \(\alpha\), of the equation \(f(x) = 0\) is \(1.9\) correct to \(1\) decimal place.

1. Use the iteration formula $$x_{n+1} = 2 - e^{kx_n^2},$$ with \(x_0 = 1.9\) and your value of \(k\), to find \(\alpha\) to \(3\) decimal places and justify the accuracy of your answer. [5]
2. Solve the equation \(f'(x) = 0\). [5]

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

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\frac{6}{\sqrt{2x-7}}\) [2]
2. \(x^2 e^{-x}\) [3]

A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

Show that the curve \(y = x^2 \ln x\) has a stationary point when \(x = \frac{1}{\sqrt{e}}\). [6]

The equation of a curve is \(y = \frac{x^2}{2x + 1}\).

1. Show that \(\frac{dy}{dx} = \frac{2x(x + 1)}{(2x + 1)^2}\). [4]
2. Find the coordinates of the stationary points of the curve. You need not determine their nature. [4]

The function \(\text{f}(x) = \frac{\sin x}{2 - \cos x}\) has domain \(-\pi \leqslant x \leqslant \pi\). Fig. 8 shows the graph of \(y = \text{f}(x)\) for \(0 \leqslant x \leqslant \pi\). \includegraphics{figure_6}

1. Find \(\text{f}(-x)\) in terms of \(\text{f}(x)\). Hence sketch the graph of \(y = \text{f}(x)\) for the complete domain \(-\pi \leqslant x \leqslant \pi\). [3]
2. Show that \(\text{f}'(x) = \frac{2\cos x - 1}{(2 - \cos x)^2}\). Hence find the exact coordinates of the turning point P. State the range of the function \(\text{f}(x)\), giving your answer exactly. [8]
3. Using the substitution \(u = 2 - \cos x\) or otherwise, find the exact value of \(\int_0^\pi \frac{\sin x}{2 - \cos x} dx\). [4]
4. Sketch the graph of \(y = \text{f}(2x)\). [1]
5. Using your answers to parts (iii) and (iv), write down the exact value of \(\int_0^{\frac{\pi}{2}} \frac{\sin 2x}{2 - \cos 2x} dx\). [2]

A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve. [7]

Fig. 8 shows part of the curve \(y = \text{f}(x)\), where \(\text{f}(x) = e^{-\frac{1}{5}x} \sin x\), for all \(x\). \includegraphics{figure_8}

1. Sketch the graphs of (A) \(y = \text{f}(2x)\), (B) \(y = \text{f}(x + \pi)\). [4]
2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P. [6]
3. Show that \(\text{f}(x + \pi) = -e^{-\frac{1}{5}\pi}\text{f}(x)\). Hence, using the substitution \(u = x - \pi\), show that $$\int_{\pi}^{2\pi} \text{f}(x)\,dx = -e^{-\frac{1}{5}\pi} \int_{0}^{\pi} \text{f}(u)\,du.$$ Interpret this result graphically. [You should not attempt to integrate f(x).] [8]

A curve is defined by the equation \(y = 2x \ln(1 + x)\).

1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]

1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]

\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Find the volume of the solid of revolution generated. [2]
2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\ln(\cos x)\) [2]
2. \(x^2 \sin 3x\) [2]

A curve, C, has equation $$y = \frac{e^{3x-5}}{x^2}$$ Show that C has exactly one stationary point. Fully justify your answer. [7 marks]

At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$

1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]

Given that $$y = \frac{x^3}{\sin x}$$ find \(\frac{dy}{dx}\) [3 marks]

A curve has equation \(y = 2x \cos 3x + (3x^2 - 4) \sin 3x\)

1. Find \(\frac{dy}{dx}\), giving your answer in the form \((mx^2 + n) \cos 3x\), where \(m\) and \(n\) are integers. [4 marks]
2. Show that the \(x\)-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 marks]

The curve \(C\) with equation $$y = (x^2 - 8x) \ln x$$ is defined for \(x > 0\) and is shown in the diagram below. \includegraphics{figure_11} The shaded region, \(R\), lies below the \(x\)-axis and is bounded by \(C\) and the \(x\)-axis. Show that the area of \(R\) can be written as $$p + q \ln 2$$ where \(p\) and \(q\) are rational numbers to be found. [10 marks]