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

f(x) = \(x + \frac{e^x}{5}\), \(x \in \mathbb{R}\).

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

The curve \(C\), with equation \(y = \)f(x), crosses the \(y\)-axis at the point \(A\).

1. Find an equation for the tangent to \(C\) at \(A\). [3]
2. Complete the table, giving the values of \(\sqrt{x + \frac{e^x}{5}}\) to 2 decimal places.

\(x\)	0	0.5	1	1.5	2
\(\sqrt{x + \frac{e^x}{5}}\)	0.45	0.91

[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]

Fig. 9 shows the curve \(y = \frac{x^2}{3x - 1}\). P is a turning point, and the curve has a vertical asymptote \(x = a\). \includegraphics{figure_1}

1. Write down the value of \(a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{x(3x - 2)}{(3x - 1)^2}\) [3]
3. Find the exact coordinates of the turning point P. Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point. [7]
4. Using the substitution \(u = 3x - 1\), show that \(\int \frac{x^2}{3x - 1} dx = \frac{1}{27} \int \left( u + 2 + \frac{1}{u} \right) du\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac{2}{3}\) and \(x = 1\). [7]

\includegraphics{figure_2} Figure 2 shows a sketch of part of two curves \(C_1\) and \(C_2\) for \(y \geq 0\). The equation of \(C_1\) is \(y = m_1 - x^{n_1}\) and the equation of \(C_2\) is \(y = m_2 - x^{n_2}\), where \(m_1\), \(m_2\), \(n_1\) and \(n_2\) are positive integers with \(m_2 > m_1\). Both \(C_1\) and \(C_2\) are symmetric about the line \(x = 0\) and they both pass through the points \((3, 0)\) and \((-3, 0)\). Given that \(n_1 + n_2 = 12\), find

1. the possible values of \(n_1\) and \(n_2\), [4]
2. the exact value of the smallest possible area between \(C_1\) and \(C_2\), simplifying your answer, [8]
3. the largest value of \(x\) for which the gradients of the two curves can be the same. Leave your answer in surd form. [5]

The cubic polynomial \(\text{f}(x)\) is defined by \(\text{f}(x) = x^3 + px + q\), where \(p\) and \(q\) are constants.

1. 1. Given that \(\text{f}'(2) = 13\), find the value of \(p\). [2]
   2. Given also that \((x - 2)\) is a factor of \(\text{f}(x)\), find the value of \(q\). [2]
   The curve \(y = \text{f}(x)\) is translated by the vector \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\).
2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]

A curve has equation \(y = 2x^2 - 8x\sqrt{x} + 8x + 1\) for \(x \geq 0\)

1. Prove that the curve has a maximum point at \((1, 3)\) Fully justify your answer. [9 marks]
2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]

Prove that the curve with equation $$y = 2x^5 + 5x^4 + 10x^3 - 8$$ has only one stationary point, stating its coordinates. [6 marks]

A curve has equation $$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$ where \(a\), \(b\) and \(c\) are positive constants. The curve has a single turning point. Use the second derivative of \(y\) to determine the nature of this turning point. You do not need to find the coordinates of the turning point. Fully justify your answer. [7 marks]

A curve has equation \(y = f(x)\) where $$f(x) = x(6 - x)$$

1. Find \(f'(x)\) [2 marks]
2. The diagram below shows the graph of \(y = f(x)\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes. [3 marks] \includegraphics{figure_9}

Chris claims that, "for any given value of \(x\), the gradient of the curve \(y = 2x^3 + 6x^2 - 12x + 3\) is always greater than the gradient of the curve \(y = 1 + 60x - 6x^2\)". Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true. [7 marks]

A curve has equation \(y = 6x\sqrt{x} + \frac{32}{x}\) for \(x > 0\)

1. Find \(\frac{dy}{dx}\) [4 marks]
2. The point \(A\) lies on the curve and has \(x\)-coordinate 4 Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis. [5 marks]

A particle, of mass 400 grams, is initially at rest at the point \(O\). The particle starts to move in a straight line so that its velocity, \(v\) m s⁻¹, at time \(t\) seconds is given by \(v = 6t^2 - 12t^3\) for \(t > 0\)

1. Find an expression, in terms of \(t\), for the force acting on the particle. [3 marks]
2. Find the time when the particle next passes through \(O\). [5 marks]

It is given that $$y = 3x^4 + \frac{2}{x} - \frac{x}{4} + 1$$ Find an expression for \(\frac{d^2y}{dx^2}\) [3 marks]

A curve has equation \(y = \frac{2}{\sqrt{x}}\) Find \(\frac{dy}{dx}\) Circle your answer. [1 mark] \(\frac{\sqrt{x}}{3}\) \quad \(\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{x\sqrt{x}}\) \quad \(-\frac{1}{2x\sqrt{x}}\)

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]

A particle is moving in a straight line through the origin \(O\) The displacement of the particle, \(r\) metres, from \(O\), at time \(t\) seconds is given by $$r = p + 2t - qe^{-0.2t}$$ where \(p\) and \(q\) are constants. When \(t = 3\), the acceleration of the particle is \(-1.8\) m s\(^{-2}\)

1. Show that \(q \approx 82\) [5 marks]
2. The particle has an initial displacement of 5 metres. Find the value of \(p\) Give your answer to two significant figures. [2 marks]

A curve has equation \(y = x^5 + 4x^3 + 7x + q\) where \(q\) is a positive constant. Find the gradient of the curve at the point where \(x = 0\) Circle your answer. [1 mark] \(0\) \quad \(4\) \quad \(7\) \quad \(q\)

A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)

1. 1. Find \(f''(x)\) [2 marks]
   2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
   1. State the single transformation which maps f onto g. [1 mark]
   2. State the range of values of \(x\) for which g is an increasing function. [1 mark]

A curve has equation \(y = x^4 + 2^x\) Find an expression for \(\frac{dy}{dx}\) [2 marks]

The curve \(C\) has equation $$y = 2x^2 - 12x + 16$$ Find the gradient of the curve at the point \(P (5, 6)\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [4]

\includegraphics{figure_4} Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool \(ABCDEA\) consists of a rectangular section \(ABDE\) joined to a semicircular section \(BCD\) as shown in Figure 4. Given that \(AE = 2x\) metres, \(ED = y\) metres and the area of the pool is \(250\text{m}^2\),

1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2x + \frac{250}{x} + \frac{\pi x}{2}$$ [4]
2. Explain why \(0 < x < \sqrt{\frac{500}{\pi}}\) [2]
3. Find the minimum perimeter of the pool, giving your answer to \(3\) significant figures. [4]

\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).

1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]

Given that \(x\) can vary,

1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]

\(f(x) = -2x^3 - x^2 + 4x + 3\)

1. Use the factor theorem to show that \((3 - 2x)\) is a factor of \(f(x)\). [2]
2. Hence show that \(f(x)\) can be written in the form \(f(x) = (3 - 2x)(x + a)^2\) where \(a\) is an integer to be found. [4]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve with equation \(y = f(x)\).

1. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(f(x) \leq 0\)
   2. \(f'(\frac{x}{2}) = 0\)
   [3]