1.07m Tangents and normals: gradient and equations

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]

In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]

A curve has equation \(x^3 - 3x^2y + y^2 + 1 = 0\).

1. Show that \(\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}\). [4]
2. Find the equation of the normal to the curve at the point \((1, 2)\). [4]

Find the equation of the tangent to the curve $$y = 3x^2(x + 2)^6$$ at the point \((-1, 3)\), giving your answer in the form \(y = mx + c\). [5]

The curve with equation \(y = f(x)\) where $$f(x) = x^2 + \ln(2x^2 - 4x + 5)$$ has a single turning point at \(x = \alpha\)

1. Show that \(\alpha\) is a solution of the equation $$2x^3 - 4x^2 + 7x - 2 = 0$$ [4]

The iterative formula $$x_{n+1} = \frac{1}{7}(2 + 4x_n^2 - 2x_n^3)$$ is used to find an approximate value for \(\alpha\). Starting with \(x_1 = 0.3\)

1. calculate, giving each answer to 4 decimal places,
   1. the value of \(x_2\)
   2. the value of \(x_4\)
   [2]

Using a suitable interval and a suitable function that should be stated,

1. show that \(\alpha\) is 0.341 to 3 decimal places. [2]

In this question you should show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x^3 - 10x^2 + 27x - 23$$ The point \(P(5, -13)\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [4]
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. [1]

The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).

1. Use algebraic integration to find the exact area of \(R\). [4]

Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]

Curve C has equation $$y = x^3 - 7x^2 + 5x + 4$$ The point \(P(2, -6)\) lies on \(C\) Find the equation of the tangent to \(C\) at \(P\) Give your answer in the form \(y = mx + c\) where \(m\) and \(c\) are integers to be found. [3]

The curve \(C\) has equation $$y = \frac{3 + \sin 2x}{2 + \cos 2x}$$

1. Show that $$\frac{dy}{dx} = \frac{6\sin 2x + 4\cos 2x + 2}{(2 + \cos 2x)^2}$$ [4]
2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac{\pi}{2}\). Write your answer in the form \(y = ax + b\), where \(a\) and \(b\) are exact constants. [4]

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]

In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4\sqrt{x - 3x + 1}\) 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. [7]

The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$

1. Find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [2]

The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).

1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]

The line \(l\) intersects the curve \(C\) again at the point \(Q\).

1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]

1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
   2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{-\frac{x}{2}}$$ [2 marks]
   3. Use the recurrence relation $$x_{n+1} = e - e^{-\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
   4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{-\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_1}

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan\alpha\). [3]

\includegraphics{figure_9} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x, \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P(e, e)\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(Ae^2 + B\) where \(A\) and \(B\) are rational numbers to be found. [9]

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]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]

1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
   2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{\frac{x}{2}}$$ [2 marks]
   3. Use the recurrence relation $$x_{n+1} = e - e^{\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
   4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_9}

A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t^2, \quad y = t^3.$$

1. Show that the equation of the tangent at the point with parameter \(t\) is $$2y = 3tx - t^3.$$ [4]

1. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A\left(\frac{19}{2}, -\frac{15}{8}\right)\) and it meets the \(x\)-axis at the point \(B\). Find the area of triangle \(OAB\), where \(O\) is the origin. [7]

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]

The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).

1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
3. 1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
   2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]

The function f is defined by \(f(x) = \sqrt{x}, x > 0\).

1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]

The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).

1. 1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
   2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]

The equation of a curve is \(y = x^{\frac{3}{2}} \ln x\). Find the exact coordinates of the stationary point on the curve. [5]

% Figure 2 shows curve with vertical asymptotes at x = -2 and x = 2, horizontal asymptote at y = 1, with U-shaped region between asymptotes \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{x^2 - 2}{x^2 - 4}\) and \(x \neq \pm 2\). The curve cuts the \(y\)-axis at \(U\).

1. Write down the coordinates of the point \(U\). [1]

The point \(P\) with \(x\)-coordinate \(a\) (\(a \neq 0\)) lies on \(C\).

1. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left(0, \frac{a^2 - 2}{a^2 - 4} - \frac{(a^2 - 4)^2}{4}\right)$$ [6]

The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).

1. 1. Show that $$\frac{a^2}{2(a^2-4)} - \frac{(a^2-4)^2}{4} = a^2 + \frac{(a^2-4)^4}{16}$$
   2. Hence, show that $$(a^2 - 4)^2 = 1$$
   3. Find the centre and radius of \(E\).
   [10]

[Total 17 marks]