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

\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)

A curve \(C\) has equation \(y = \frac{1}{9}x^3 - kx + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient \(-9\). The \(x\)-coordinate of \(Q\) is \(3\).

1. Show that \(k = 12\). [3]
2. Find the coordinates of each of the stationary points of \(C\) and determine their nature. [6]
3. Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. [2]

Given that \(y = 12\sqrt{x} - \frac{27}{x} + 4\), find the value of \(\frac{dy}{dx}\) when \(x = 9\). [4]

A curve has equation $$y = 2x^3 - 2x^2 - 2x + 8$$

1. Find \(\frac{dy}{dx}\) [2]
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation. [4]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]

A curve has equation \(y = \frac{1}{4}x^4 - x^3 - 2x^2\).

1. Find \(\frac{dy}{dx}\). [1]
2. Hence sketch the gradient function for the curve. [4]
3. Find the equation of the tangent to the curve \(y = \frac{1}{4}x^4 - x^3 - 2x^2\) at \(x = 4\). [2]

A curve has equation $$y = 2x^3 - 4x + 5$$ Find the equation of the tangent to the curve at the point \(P(2, 13)\). Write your answer in the form \(y = mx + c\), where \(m\) and \(c\) are integers to be found. Solutions relying on calculator technology are not acceptable. [5]

A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$

1. Find \(\frac{dy}{dx}\). [2 marks]
2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]

A particle \(P\) moves in a straight line. At time \(t\) s the displacement \(s\) cm from a fixed point \(O\) is given by: $$s = \frac{1}{6}\left(8t^3 - 105t^2 + 144t + 540\right).$$ Find the distance between the points at which the particle is instantaneously at rest. [7]

A car starts from the point \(A\). At time \(t\) s after leaving \(A\), the distance of the car from \(A\) is \(s\) m, where \(s = 30t - 0.4t^2\), \(0 \leq t \leq 25\). The car reaches the point \(B\) when \(t = 25\).

1. Find the distance \(AB\). [2]
2. Show that the car travels with a constant acceleration and state the value of this acceleration. [3]

A runner passes through \(B\) when \(t = 0\) with an initial velocity of \(2 \text{ m s}^{-1}\) running directly towards \(A\). The runner has a constant acceleration of \(0.1 \text{ m s}^{-2}\).

1. Find the distance from \(A\) at which the runner and the car pass one another. [8]

\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$

1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]

The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).

1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]

In this question you must show detailed reasoning Find the equation of the normal to the curve \(y = \frac{x^2-32}{\sqrt{x}}\) at the point on the curve where \(x = 4\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

The cubic polynomial \(2x^3 - kx^2 + 4x + k\), where \(k\) is a constant, is denoted by f(x). It is given that f'(2) = 16.

1. Show that \(k = 3\). [3]

For the remainder of the question, you should use this value of \(k\).

1. Use the factor theorem to show that \((2x + 1)\) is a factor of f(x). [2]
2. Hence show that the equation f(x) = 0 has only one real root. [3]

The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$

1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
2. Find the nature of the stationary point, justifying your answer. [2]

The curve \(C\) has equation $$y = \frac{1}{2}x^3 - 9x^2 + \frac{8}{x} + 30, \quad x > 0$$

1. Find \(\frac{dy}{dx}\). [4]
2. Show that the point \(P(4, -8)\) lies on \(C\). [2]
3. Find an equation of the normal to \(C\) at the point \(P\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]

In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac{2}{3}x^3 + \frac{5}{2}x^2 - 3x + 7\) is positive. Give your answer in set notation. [5]

A curve has equation \(y = x^5 - 5x^4\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Verify that the curve has a stationary point when \(x = 4\). [2]
3. Determine the nature of this stationary point. [2]

A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant. The point \(P\) on the curve has \(x\)-coordinate 4. The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(PQ\) is the radius of a circle centre \(P\). Show that \(k = \frac{1}{2}\). Find the equation of the circle. [10]

A student is attempting to model the flight of a boomerang. She throws the boomerang from a fixed point \(O\) and catches it when it returns to \(O\). She suggests the model for the displacement, \(s\) metres, after \(t\) seconds is given by \(s = 9t^2 - \frac{3}{2}t^3\), \(0 \leq t \leq 6\). For this model,

1. determine what happens at \(t = 6\), [2]
2. find the greatest displacement of the boomerang from \(O\), [4]
3. find the velocity of the boomerang 1 second before the student catches it, [2]
4. find the acceleration of the boomerang 1 second before the student catches it. [2]

\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and perimeter, \(P\), of the sector. [2]
2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]