Tangents, normals and gradients

Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]

3 A curve has equation \(y = 2 x ^ { 5 } + 5 x ^ { 4 } - 1\).

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. The point on the curve where \(x = - 1\) is \(P\).
   1. Determine whether \(y\) is increasing or decreasing at \(P\), giving a reason for your answer.
   2. Find an equation of the tangent to the curve at \(P\).
3. The point \(Q ( - 2,15 )\) also lies on the curve. Verify that \(Q\) is a maximum point of the curve.
   [0pt] [4 marks]

18 A student is investigating the intersection points of tangents to the curve \(y = 6 x ^ { 2 } - 7 x + 1\). She uses software to draw tangents at pairs of points with \(x\)-coordinates differing by 5 . Find the equation of the curve that all the intersection points lie on.

6. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\).
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.

8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P ( 4,8 )\) lies on \(C\).
3. Show that an equation of the normal to \(C\) at the point \(P\) is $$3 y = x + 20 .$$ The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
4. Find the length \(P Q\), giving your answer in a simplified surd form.

\includegraphics{figure_3} The curve \(C_1\), shown in Figure 3, has equation \(y = 4x^2 - 6x + 4\). The point \(P\left(\frac{1}{2}, 2\right)\) lies on \(C_1\) The curve \(C_2\), also shown in Figure 3, has equation \(y = \frac{1}{2}x + \ln(2x)\). The normal to \(C_1\) at the point \(P\) meets \(C_2\) at the point \(Q\). Find the exact coordinates of \(Q\). (Solutions based entirely on graphical or numerical methods are not acceptable.) [8]

Find the equation of the normal to the curve \(y = 8x^4 + 4\) at the point where \(x = \frac{1}{2}\). [5]

At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer. [1 mark] \(-10\) \quad \(-0.1\) \quad \(0.1\) \quad \(10\)

7 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).

1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the inequalities that define the shaded region, including its boundaries.

Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [6]

1. Given

$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$

1. find \(\int y \mathrm {~d} x\), simplifying each term.
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

1. Show that, for all non-zero values of the constant \(k\), the curve $$y = \frac{kx^2 - 1}{kx^2 + 1}$$ has exactly one stationary point. [5]
2. Show that, for all non-zero values of the constant \(m\), the curve $$y = e^{mx}(x^2 + mx)$$ has exactly two stationary points. [7]

1 Differentiate \(10 x ^ { 4 } + 12\).

2 Given that \(y = 2 x ^ { 3 }\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Circle your answer.
[0pt] [1 mark] \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 4 } } { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 3 }\)

7. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation \(y = 2 x ^ { 2 } + 5\) The point \(P ( 2,13 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(2 + h\) also lies on the curve.
2. Find, in terms of \(h\), the gradient of the line \(P Q\). Give your answer in simplest form.
3. Explain briefly the relationship between the answer to (b) and the answer to (a).

The curve \(y = \frac{10}{2x + 1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

1. Show that the equation of \(AC\) is \(5y + 4x = 8\). [5]
2. Find the distance \(AC\). [2]

7. The curve \(C\) has equation \(y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0\). The point \(P\) on \(C\) has \(x\)-coordinate 1 .

1. Show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\) is 3 .
2. Find an equation of the tangent to \(C\) at \(P\). This tangent meets the \(x\)-axis at the point \(( k , 0 )\).
3. Find the value of \(k\).

6 \includegraphics[max width=\textwidth, alt={}, center]{fc7679bf-a9a1-493d-bf89-35206382787f-3_576_821_258_662} The diagram shows the curve with equation \(y = \sqrt { 3 x - 5 }\). The tangent to the curve at the point \(P\) passes through the origin. The shaded region is bounded by the curve, the \(x\)-axis and the line \(O P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 10 } { 3 }\) and hence find the exact area of the shaded region.

1 Differentiate \(6 x ^ { \frac { 5 } { 2 } } + 4\).

1. Given that

$$y = 4 x ^ { 3 } - 1 + 2 x ^ { \frac { 1 } { 2 } } , \quad x > 0 ,$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). \includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-02_29_45_2690_1852}

3. A curve has equation $$y = \sqrt { 2 } x ^ { 2 } - 6 \sqrt { x } + 4 \sqrt { 2 } , \quad x > 0$$ Find the gradient of the curve at the point \(P ( 2,2 \sqrt { 2 } )\).
Write your answer in the form \(a \sqrt { 2 }\), where \(a\) is a constant.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \(L\)

Find the coordinates of the points on the curve \(y = \frac{1}{3}x^3 + \frac{9}{x}\) at which the tangent is parallel to the line \(y = 8x + 3\). [10]

9. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(C\) with equation $$y = ( x - 1 ) \left( x ^ { 2 } - 4 \right) .$$ The curve cuts the \(x\)-axis at the points \(P , ( 1,0 )\) and \(Q\), as shown in Figure 2.

1. Write down the \(x\)-coordinate of \(P\), and the \(x\)-coordinate of \(Q\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 2 x - 4\).
3. Show that \(y = x + 7\) is an equation of the tangent to \(C\) at the point ( \(- 1,6\) ). The tangent to \(C\) at the point \(R\) is parallel to the tangent at the point ( \(- 1,6\) ).
4. Find the exact coordinates of \(R\).

11 In this question you must show detailed reasoning. Fig. 11 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic function. Fig. 11 also shows the coordinates of the turning points and the points of intersection with the axes. \begin{figure}[h] \end{figure} Show that the tangent to \(y = \mathrm { f } ( x )\) at \(x = t\) is parallel to the tangent to \(y = \mathrm { f } ( x )\) at \(x = - t\) for all values of \(t\).

Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]

9. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      [0pt] [2 marks]
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
      [0pt] [2 marks]
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

\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]

It is given that \(f(x) = \frac{6}{x^2} + 2x\).

1. Find \(f'(x)\). [3]
2. Find \(f''(x)\). [2]

1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find

1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]

2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h] \end{figure}

\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]

A solid rectangular block has a square base of side \(x\) cm. The height of the block is \(h\) cm and the total surface area of the block is \(96\) cm\(^2\).

1. Express \(h\) in terms of \(x\) and show that the volume, \(V\) cm\(^3\), of the block is given by $$V = 24x - \frac{1}{2}x^3.$$ [3]

Given that \(x\) can vary,

1. find the stationary value of \(V\), [3]
2. determine whether this stationary value is a maximum or a minimum. [2]

5 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667} 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 \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
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.

A curve with equation \(y = f(x)\) passes through the points \((0, 2)\) and \((3, -1)\). It is given that \(f'(x) = kx^2 - 2x\), where \(k\) is a constant. Find the value of \(k\). [5]

3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$

A curve is such that \(\frac{d^2y}{dx^2} = -4x\). The curve has a maximum point at \((2, 12)\).

1. Find the equation of the curve. [6]

A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.

1. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing. [2]

Find the equation of the tangent to the curve \(y = \frac{2x + 1}{3x - 1}\) at the point \((1, \frac{3}{2})\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

Find the equation of the tangent to the curve \(y = \sqrt{4x + 1}\) at the point \((2, 3)\). [5]

Water is being emptied out of a sink. The depth of water, \(y\)cm, at time \(t\) seconds, may be modelled by $$y = t^2 - 14t + 49 \quad\quad 0 \leqslant t \leqslant 7.$$

1. Find the value of \(t\) when the depth of water is 25cm. [3]
2. Find the rate of decrease of the depth of water when \(t = 3\). [3]

1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]