1.07a Derivative as gradient: of tangent to curve

5 Fig. 5.1 shows part of the curve \(y = x ^ { \frac { 1 } { 2 } }\). P is the point \(( 1,1 )\) and \(Q\) is the point on the curve with \(x\)-coordinate \(1 + h\). \begin{figure}[h] \end{figure} Table 5.2 shows, for different values of \(h\), the coordinates of P , the coordinates of Q , the change in \(y\) from P to Q and the gradient of the chord PQ . \begin{table}[h] \end{table}

1. Fill in the missing values for the case \(h = 1\) in the copy of Table 5.2 in the Printed Answer Booklet. Give your answers correct to 6 decimal places where necessary.
2. Explain how the sequence of values in the last column of Table 5.2 relates to the gradient of the curve \(y = x ^ { \frac { 1 } { 2 } }\) at the point \(P\).
3. Use calculus to find the gradient of the curve at the point P .

4 The diagram shows points \(A\) and \(B\) on the curve \(y = \left( \frac { x } { 4 } \right) ^ { - x }\).
The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1 . \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-4_522_707_1758_278}

1. Find the gradient of chord AB . Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A .

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).

6

1. 1. Using the binomial expansion, or otherwise, express \(( 2 + x ) ^ { 3 }\) in the form \(8 + a x + b x ^ { 2 } + x ^ { 3 }\), where \(a\) and \(b\) are integers. (3 marks)
   2. Write down the expansion of \(( 2 - x ) ^ { 3 }\).
2. Hence show that \(( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 } = 24 x + 2 x ^ { 3 }\).
3. Hence show that the curve with equation $$y = ( 2 + x ) ^ { 3 } - ( 2 - x ) ^ { 3 }$$ has no stationary points.

2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .

3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. State the range of values of the acceleration of the particle.
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

2 A curve has equation $$y = x ^ { 2 } - 6 x + 5$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates 2 and \(2 + h\) respectively.

1. Find, in terms of \(h\), the gradient of the line \(A B\), giving your answer in its simplest form.
2. Explain how the result of part (a) can be used to find the gradient of the curve at \(A\). State the value of this gradient.

5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).

1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
   \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}

5

1. A curve has equation \(y = 2 x ^ { 2 } - 5 x\).
   The point \(P\) on the curve has coordinates \(( 1 , - 3 )\).
   The point \(Q\) on the curve has \(x\)-coordinate \(1 + h\).
   1. Show that the gradient of the line \(P Q\) is \(2 h - 1\).
   2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point \(P\) is parallel to the line \(x + y = 0\).
2. For the improper integral \(\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x\), either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.

4 Fig. 4 shows the graph of \(y = x ^ { 5 } - 6 \sqrt { x } + 4\). \begin{figure}[h] \end{figure} There are two roots of the equation \(x ^ { 5 } - 6 \sqrt { x } + 4 = 0\). The roots are \(\alpha\) and \(\beta\), such that \(\alpha < \beta\).

1. Show that \(0 < \alpha < 1\) and \(1 < \beta < 2\).
2. Obtain the Newton-Raphson iterative formula $$x _ { n + 1 } = x _ { n } - \frac { x _ { n } ^ { \frac { 11 } { 2 } } - 6 x _ { n } + 4 \sqrt { x _ { n } } } { 5 x _ { n } ^ { \frac { 9 } { 2 } } - 3 }$$
3. Use the iterative formula found in part (b) with a starting value of \(x _ { 0 } = 1\) to obtain \(\beta\) correct to 6 decimal places.
4. Use the iterative formula found in part (b) with a starting value of \(x _ { 0 } = 0\) to find \(x _ { 1 }\).
5. Give a geometrical explanation of why the Newton-Raphson iteration fails to find \(\alpha\) in part (d).
6. Obtain the iterative formula $$x _ { n + 1 } = \left( \frac { x _ { n } ^ { 5 } + 4 } { 6 } \right) ^ { 2 }$$
7. Use the iterative formula found in part (f) with a starting value of \(x _ { 0 } = 0\) to obtain \(\alpha\) correct to 6 decimal places.

4 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Calculate four estimates of the derivative of \(\mathrm { f } ( x )\) at \(x = 4\).
2. Without doing any further calculation, state the value of \(f ^ { \prime } ( 4 )\) as accurately as you can, justifying the precision quoted.

4 The table below gives values of a function \(y = \mathrm { f } ( x )\).

1. Calculate three estimates of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) using the central difference method.
2. State the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) to an appropriate degree of accuracy. Justify your answer.

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}

1. 1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
   2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
   3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
   1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
   2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.

15 A curve has equation \(y = x ^ { 3 } - 48 x\) The point \(A\) on the curve has \(x\) coordinate - 4
The point \(B\) on the curve has \(x\) coordinate \(- 4 + h\) 15

1. Show that the gradient of the line \(A B\) is \(h ^ { 2 } - 12 h\) 15
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-25_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-26_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-27_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-28_2498_1721_213_150}

10 The function h is defined by $$\mathrm { h } ( x ) = \frac { \sqrt { x } } { x - 3 }$$ where \(h\) has its maximum possible domain.
10

1. Find the domain of h .
   Give your answer using set notation. 10
2. Alice correctly calculates $$h ( 1 ) = - 0.5 \text { and } h ( 4 ) = 2$$ She then argues that since there is a change of sign there must be a value of \(x\) in the interval \(1 < x < 4\) that gives \(\mathrm { h } ( x ) = 0\) Explain the error in Alice's argument.
   [0pt] [2 marks]
   10
3. By considering any turning points of h , determine whether h has an inverse function. Fully justify your answer.
   [0pt] [6 marks]

1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.

The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$

1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.

11 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 4 x - 7 = 0\) has a single root \(\alpha\), close to 1.9 , which can be found using an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\). Three possible functions that can be used for such an iteration are $$\mathrm { F } _ { 1 } ( x ) = \frac { 7 } { 4 } + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } , \quad \mathrm {~F} _ { 2 } ( x ) = \sqrt [ 3 ] { 2 x ^ { 2 } - 4 x + 7 } , \quad \mathrm {~F} _ { 3 } ( x ) = \frac { 7 - 4 x } { x ^ { 2 } - 2 x }$$

1. Differentiate each of these functions with respect to \(x\).
2. Without performing any iterations, and using \(x = 1.9\), show that an iterative process based on only two of the given functions will converge. Determine which one will do so more rapidly. The sequence of errors, \(e _ { n }\), is such that \(e _ { n + 1 } \approx \mathrm {~F} ^ { \prime } ( \alpha ) e _ { n }\).
3. Using the iteration from part (ii) with the most rapid convergence, estimate the number of iterations required to reduce the magnitude of the error from \(\left| e _ { 1 } \right|\) in the first term to less than \(10 ^ { - 10 } \left| e _ { 1 } \right|\).

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

5 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-3_565_730_219_669} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) 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 the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)

1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]

The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).

1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
2. Deduce the value of f\('(2)\) using the values in the table. [1]

The equation of a curve is \(y = 2x^2 - 3\). Two points \(A\) and \(B\) with \(x\)-coordinates 2 and \((2 + h)\) respectively lie on the curve.

1. Find and simplify an expression for the gradient of the chord \(AB\) in terms of \(h\). [3]
2. Explain how the gradient of the curve at the point \(A\) can be deduced from the answer to part (a), and state the value of this gradient. [2]

A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\) is given by $$v = 12t - 4t^2 \quad \text{for } 0 \leqslant t \leqslant 2,$$ $$v = 16 - 4t \quad \text{for } 2 \leqslant t \leqslant 4.$$

1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]