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

1 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt { 4 - x ^ { 2 } }\) for \(- 2 \leqslant x \leqslant 2\).

1. Show that the curve \(y = \sqrt { 4 - x ^ { 2 } }\) is a semicircle of radius 2 , and explain why it is not the whole of this circle. Fig. 9 shows a point \(\mathrm { P } ( a , b )\) on the semicircle. The tangent at P is shown. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9e68f5e0-3394-4962-acd9-25bb31f09f2b-1_628_935_728_657} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
2. (A) Use the gradient of OP to find the gradient of the tangent at P in terms of \(a\) and \(b\).
   (B) Differentiate \(\sqrt { 4 - x ^ { 2 } }\) and deduce the value of \(\mathrm { f } ^ { \prime } ( a )\).
   (C) Show that your answers to parts (A) and (B) are equivalent. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } ( x - 2 )\), for \(0 \leqslant x \leqslant 4\).
3. Describe a sequence of two transformations that would map the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\). Hence sketch the curve \(y = \mathrm { g } ( x )\).
4. Show that if \(y = \mathrm { g } ( x )\) then \(9 x ^ { 2 } + y ^ { 2 } = 36 x\).

2 Fig. 7 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x \sqrt { 1 + x }\). The curve meets the \(x\)-axis at the origin and at the point P . \begin{figure}[h] \end{figure}

1. Verify that the point P has coordinates \(( - 1,0 )\). Hence state the domain of the function \(\mathrm { f } ( x )\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + 3 x } { 2 \sqrt { 1 + x } }\).
3. Find the exact coordinates of the turning point of the curve. Hence write down the range of the function.
4. Use the substitution \(u = 1 + x\) to show that $$\int _ { - 1 } ^ { 0 } x \sqrt { 1 + x } \mathrm {~d} x = \int _ { 0 } ^ { 1 } \left( \begin{array} { l l } u ^ { \frac { 3 } { 2 } } & u ^ { \frac { 1 } { 2 } } \end{array} \right) \mathrm { d } u .$$ Hence find the area of the region enclosed by the curve and the \(x\)-axis.

1 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence find \(\mathrm { f } ^ { \prime } ( x )\) and deduce that the gradient of the curve is negative at all points on the curve.

2 It is given that \(\mathrm { f } ( x ) = x ^ { 2 } - \tan ^ { - 1 } x\).

1. Show by calculation that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval \(0.8 < x < 0.9\).
2. Use the Newton-Raphson method, with a first approximation 0.8, to find the next approximation to this root. Give your answer correct to 3 decimal places.

7. Differentiate with respect to \(x\), giving each answer in its simplest form.

1. \(( 1 - 2 x ) ^ { 2 }\)
2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).

1. On separate diagrams, sketch the curve in each of the following cases.
   (A) \(m = 1 , n = 1\),
   (B) \(m = 2 , n = 2\),
   (C) \(m = 2 , n = 4\),
   (D) \(m = 4 , n = 2\).
2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. }{www.ocr.org.uk}) after the live examination series.
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5 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = 10 x ^ { - 5 }\),
2. \(y = \sqrt [ 4 ] { x }\),
3. \(y = x ( x + 3 ) ( 1 - 5 x )\).

9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.

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

1. Find the gradient of the curve at the point for which \(x = 2\).
2. Find the equation of the normal to the curve at the point for which \(x = 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Find the values of \(k\) for which the line \(y = k x - 4\) is a tangent to the curve.

3 Find the equation of the normal to the curve \(y = x ^ { 3 } - 4 x ^ { 2 } + 7\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 \includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_394_846_1868_648} The diagram shows part of the curve \(y = x ^ { 2 } + 5\). The point \(A\) has coordinates ( 1,6 ). The point \(B\) has coordinates ( \(a , a ^ { 2 } + 5\) ), where \(a\) is a constant greater than 1 . The point \(C\) is on the curve between \(A\) and \(B\).

1. Find by differentiation the value of the gradient of the curve at the point \(A\).
2. The line segment joining the points \(A\) and \(B\) has gradient 2.3. Find the value of \(a\).
3. State a possible value for the gradient of the line segment joining the points \(A\) and \(C\).

9 Given that \(\mathrm { f } ( x ) = \frac { 1 } { x } - \sqrt { x } + 3\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } ( 4 )\).

6 Given that \(y = \frac { 5 } { x ^ { 2 } } - \frac { 1 } { 4 x } + x\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

6 Given that \(\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)\).

1 Given that \(y = x ^ { 5 } + \frac { 1 } { x ^ { 2 } }\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).

1. Find the value of \(k\).
2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT

6 Find the gradient of the curve \(y = 2 x + \frac { 6 } { \sqrt { x } }\) at the point where \(x = 4\).

7

1. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )\), find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the gradient of the curve \(y = x ^ { - \frac { 1 } { 3 } }\) at the point where \(x = - 8\).

11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .

7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).

1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
2. Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\). \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\).
3. Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
4. Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.

7 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\).

10 Fig. 10 shows a sketch of the graph of \(y = 7 x - x ^ { 2 } - 6\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at the point on the curve where \(x = 2\). Show that this tangent crosses the \(x\)-axis where \(x = \frac { 2 } { 3 }\).
2. Show that the curve crosses the \(x\)-axis where \(x = 1\) and find the \(x\)-coordinate of the other point of intersection of the curve with the \(x\)-axis.
3. Find \(\int _ { 1 } ^ { 2 } \left( 7 x - x ^ { 2 } - 6 \right) \mathrm { d } x\). Hence find the area of the region bounded by the curve, the tangent and the \(x\)-axis, shown shaded on Fig. 10.

10 \begin{figure}[h] \end{figure} A is the point with coordinates \(( 1,4 )\) on the curve \(y = 4 x ^ { 2 }\). B is the point with coordinates \(( 0,1 )\), as shown in Fig. 10.

1. The line through A and B intersects the curve again at the point C . Show that the coordinates of C are \(\left( - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)\).
2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = - 2 x - \frac { 1 } { 4 }\).
3. The two tangents intersect at the point D . Find the \(y\)-coordinate of D . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-4_773_1027_255_557} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).

3 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.