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

6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).

1. Write down the equations of the asymptotes of \(S\).
2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
3. Sketch \(S\).

2 The equation of a curve is \(y = x ^ { 3 } - 2 x ^ { 2 } - 4 x + 3\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$ a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\). The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
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0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\). Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.

1. Find \(\frac{dy}{dx}\). [2]
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]

\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.

1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]

The curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\). \begin{enumerate}[label=(\alph*)] \item Find the coordinates of \(A\) and \(B\). [4] \end enumerate}

\includegraphics{figure_6} The diagram shows the curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) and the line \(AB\). It is given that the equation of \(AB\) is \(y = \frac{2x-32}{3}\). Find the area of the shaded region between the curve and the line. [5]

The equation of a curve is \(y = f(x)\), where \(f(x) = (4x - 3)^{\frac{3}{4}} - \frac{20}{3}x\).

1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature. [6]
2. State the set of values for which the function f is increasing. [1]

The curve \(y = x^2 - \frac{a}{x}\) has a stationary point at \((-3, b)\). Find the values of the constants \(a\) and \(b\). [4]

\includegraphics{figure_7} The diagram shows part of the curve with equation \(y = \frac{12}{\sqrt{2x+1}}\). The point \(A\) on the curve has coordinates \(\left(\frac{7}{2}, 6\right)\).

1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = mx + c\). [4]
2. Find the area of the region bounded by the curve and the lines \(x = 0\), \(x = \frac{7}{2}\) and \(y = 0\). [4]

The equation of a curve is \(y = 4 + 5x + 6x^2 - 3x^3\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases. [4]
2. It is given that \(y = 9x + k\) is a tangent to the curve. Find the value of the constant \(k\). [4]

A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).

1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
2. Find f\((x)\). [4]

It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$

1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]

The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.

1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]

The equation of a curve is \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\).

1. Obtain an expression for \(\frac{dy}{dx}\). [3]
2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of \(0.12\) units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\). [2]

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

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
2. Find the coordinates of \(M\). [3]
3. Find, showing all necessary working, the area of the shaded region. [4]

A curve with equation \(y = \mathrm{f}(x)\) passes through the point \(A(3, 1)\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}\). Find the \(y\)-coordinate of \(B\). [6]

\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

The equation of a curve is \(y = \frac{1+x}{1+2x}\) for \(x > -\frac{1}{2}\). Show that the gradient of the curve is always negative. [3]

A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$

1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
2. Find the times when the particle is at rest. [2]

A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O\) 100 s later. The velocity of \(P\) is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving \(O\), where $$v = 0.0001t^3 - 0.015t^2 + 0.5t.$$

1. Show that \(P\) is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\). [2]
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocity-time graph for \(P\)'s motion for \(0 \leq t \leq 100\). [7]
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leq t \leq 100\). [4]

A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \text{ s}\) after leaving \(O\), the displacement \(s \text{ m}\) from \(O\) is given by \(s = t^3 - 4t^2 + 4t\) and the velocity is \(v \text{ m s}^{-1}\).

1. Find an expression for \(v\) in terms of \(t\). [2]
2. Find the two values of \(t\) for which \(P\) is at instantaneous rest. [2]
3. Find the minimum velocity of \(P\). [3]

A particle moves in a straight line. The displacement of the particle at time \(t\) s is \(s\) m, where $$s = t^3 - 6t^2 + 4t.$$ Find the velocity of the particle at the instant when its acceleration is zero. [4]

A particle \(P\) moves in a straight line, starting from a point \(O\). The velocity of \(P\), measured in m s\(^{-1}\), at time \(t\) s after leaving \(O\) is given by $$v = 0.6t - 0.03t^2.$$

1. Verify that, when \(t = 5\), the particle is 6.25 m from \(O\). Find the acceleration of the particle at this time. [4]
2. Find the values of \(t\) at which the particle is travelling at half of its maximum velocity. [6]

Given that \(y = 4x^3 - \frac{5}{x^2}\), \(x \neq 0\), find in their simplest form

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