1.07m Tangents and normals: gradient and equations

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]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = (3x - 1)^4\). The point P on the curve has coordinates \((1, 16)\) and the tangent to the curve at P meets the \(x\)-axis at the point Q. The shaded region is bounded by PQ, the \(x\)-axis and that part of the curve for which \(\frac{1}{3} \leqslant x \leqslant 1\). Find the exact area of this shaded region. [10]

Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x+4}}\) and the line \(x = 5\). The curve has an asymptote \(l\). The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q. \includegraphics{figure_8}

1. Show that for this curve \(\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}\). [5]
2. Find the coordinates of the point P. [4]
3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\). [9]

The curve \(C\) has the equation \(y = 2e^x - 6 \ln x\) and passes through the point \(P\) with \(x\)-coordinate \(1\).

1. Find an equation for the tangent to \(C\) at \(P\). [4]

The tangent to \(C\) at \(P\) meets the coordinate axes at the points \(Q\) and \(R\).

1. Show that the area of triangle \(OQR\), where \(O\) is the origin, is \(\frac{9}{3-e}\). [4]

A curve has the equation \(y = (2x + 3)e^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies in the interval \([-2, -1]\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3e^{x_n}}{e^{x_n} - 2}$$ with \(x_0 = -1\), to find \(x_1\), \(x_2\), \(x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [3]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

The function f is defined by $$\text{f}(x) \equiv 2 + \ln (3x - 2), \quad x \in \mathbb{R}, \quad x > \frac{2}{3}.$$

1. Find the exact value of \(\text{f}(1)\). [2]
2. Find an equation for the tangent to the curve \(y = \text{f}(x)\) at the point where \(x = 1\). [4]
3. Find an expression for \(\text{f}^{-1}(x)\). [2]

$$\text{f}(x) = e^{3x + 1} - 2, \quad x \in \mathbb{R}.$$

1. State the range of f. [1]

The curve \(y = \text{f}(x)\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).

1. Find the exact coordinates of \(P\) and \(Q\). [3]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 3ex + e - 2.$$ [4]
3. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\). [4]

The curve \(C\) has the equation \(y = x^2 - 5x + 2\ln \frac{x}{3}\), \(x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3x + 5y + 21 = 0.$$ [5]
2. Find the \(x\)-coordinates of the stationary points of \(C\). [3]

A curve has the equation \(y = (2x + 3)\mathrm{e}^{-x}\).

1. Find the exact coordinates of the stationary point of the curve. [4]

The curve crosses the \(y\)-axis at the point \(P\).

1. Find an equation for the normal to the curve at \(P\). [2]

The normal to the curve at \(P\) meets the curve again at \(Q\).

1. Show that the \(x\)-coordinate of \(Q\) lies between \(-2\) and \(-1\). [3]
2. Use the iterative formula $$x_{n+1} = \frac{3 - 3\mathrm{e}^{x_n}}{\mathrm{e}^{x_n} - 2},$$ with \(x_0 = -1\), to find \(x_1, x_2, x_3\) and \(x_4\). Give the value of \(x_4\) to 2 decimal places. [2]
3. Show that your value for \(x_4\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places. [2]

\includegraphics{figure_3} The curve \(C\) with equation \(y = 2e^x + 5\) meets the \(y\)-axis at the point \(M\), as shown in Fig. 3.

1. Find the equation of the normal to \(C\) at \(M\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

This normal to \(C\) at \(M\) crosses the \(x\)-axis at the point \(N(n, 0)\).

1. Show that \(n = 14\). [1]

The point \(P(\ln 4, 13)\) lies on \(C\). The finite region \(R\) is bounded by \(C\), the axes and the line \(PN\), as shown in Fig. 3.

1. Find the area of \(R\), giving your answers in the form \(p + q \ln 2\), where \(p\) and \(q\) are integers to be found. [7]

A curve has parametric equations $$x = 3 \cos^2 t, \quad y = \sin 2t, \quad 0 \leq t < \pi.$$

1. Show that \(\frac{dy}{dx} = -\frac{2}{3} \cot 2t\). [4]
2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis. [3]
3. Show that the tangent to the curve at the point where \(t = \frac{\pi}{6}\) has the equation $$2x + 3\sqrt{3} y = 9.$$ [3]
4. Find a cartesian equation for the curve in the form \(y^2 = \text{f}(x)\). [4]

A curve has parametric equations $$x = \frac{t}{2-t}, \quad y = \frac{1}{1+t}, \quad -1 < t < 2.$$

1. Show that \(\frac{dy}{dx} = -\frac{1}{2}\left(\frac{2-t}{1+t}\right)^2\). [4]
2. Find an equation for the normal to the curve at the point where \(t = 1\). [3]
3. Show that the cartesian equation of the curve can be written in the form $$y = \frac{1+x}{1+3x}.$$ [4]

A curve has the equation $$x^2 - 3xy - y^2 = 12.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]
2. Find an equation for the tangent to the curve at the point \((2, -2)\). [3]

A curve has the equation $$2 \sin 2x - \tan y = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [5]
2. Show that the tangent to the curve at the point \(\left(\frac{\pi}{6}, \frac{\pi}{3}\right)\) has the equation $$y = \frac{1}{2}x + \frac{\pi}{4}.$$ [3]

A curve has the equation $$3x^2 + xy - 2y^2 + 25 = 0.$$ Find an equation for the normal to the curve at the point with coordinates \((1, 4)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

A curve has the equation $$x^2 - 4xy + 2y^2 = 1.$$

1. Find an expression for \(\frac{dy}{dx}\) in its simplest form in terms of \(x\) and \(y\). [4]
2. Show that the tangent to the curve at the point \(P(1, 2)\) has the equation $$3x - 2y + 1 = 0.$$ [3]

The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).

1. Find the coordinates of \(Q\). [4]

A curve has the equation $$x^2 + 3xy - 2y^2 + 17 = 0.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. Find an equation for the normal to the curve at the point \((3, -2)\). [3]

A curve has parametric equations $$x = t^3 + 1, \quad y = \frac{2}{t}, \quad t \neq 0.$$

1. Find an equation for the normal to the curve at the point where \(t = 1\), giving your answer in the form \(y = mx + c\). [6]
2. Find a cartesian equation for the curve in the form \(y = f(x)\). [3]

A curve has parametric equations $$x = \cos 2t, \quad y = \cosec t, \quad 0 < t < \frac{\pi}{2}.$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the value of the parameter \(t\) at \(P\). [2]
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2x + 1.$$ [5]

The curve \(C\) has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of \(t\) at the points where the normal to \(C\) at \((14, 1)\) cuts \(C\) again. [11]

The points \((x, y)\) on the curve \(C\) satisfy \((x + 1)(x + 2) \frac{dy}{dx} = xy\). The line with equation \(y = 2x + 5\) is the tangent to \(C\) at a point \(P\).

1. Find the coordinates of \(P\). [4]
2. Find the equation of \(C\), giving your answer in the form \(y = f(x)\). [8]

$$f(x) = \frac{ax + b}{x + 2}; \quad x \in \mathbb{R}, x \neq -2,$$ where \(a\) and \(b\) are constants and \(b > 0\).

1. Find \(f^{-1}(x)\). [2]
2. Hence, or otherwise, find the value of \(a\) so that \(f(x) = x\). [2]

The curve \(C\) has equation \(y = f(x)\) and \(f(x)\) satisfies \(f(x) = x\).

1. On separate axes sketch
   1. \(y = f(x)\), [3]
   2. \(y = f(x - 2) + 2\). [3]

On each sketch you should indicate the equations of any asymptotes and the coordinates, in terms of \(b\), of any intersections with the axes. The normal to \(C\) at the point \(P\) has equation \(y = 4x - 39\). The normal to \(C\) at the point \(Q\) has equation \(y = 4x + k\), where \(k\) is a constant.

1. By considering the images of the normals to \(C\) on the curve with equation \(y = f(x - 2) + 2\), or otherwise, find the value of \(k\). [5]

A circle \(C\) has equation \(x^2 + y^2 - 6x + 10y + k = 0\).

1. Find the set of possible values of \(k\). [2]
2. It is given that \(k = -46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac{1}{2}\). [5]

A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\) Find the gradient of the tangent to this circle at the origin. Circle your answer. [1 mark] \(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)

A curve cuts the \(x\)-axis at \((2, 0)\) and has gradient function $$\frac{dy}{dx} = \frac{24}{x^3}$$

1. Find the equation of the curve. [4 marks]
2. Show that the perpendicular bisector of the line joining \(A(-2, 8)\) to \(B(-6, -4)\) is the normal to the curve at \((2, 0)\) [6 marks]