1.07m Tangents and normals: gradient and equations

\includegraphics{figure_2} Figure 2 shows part of the curve \(C\) with equation \(y = f(x)\), where $$f(x) = 0.5e^x - x^2.$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).

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

The \(x\)-coordinate of \(B\) is approximately \(2.15\). A more exact estimate is to be made of this coordinate using iterations \(x_{n+1} = \ln g(x_n)\).

1. Show that a possible form for \(g(x)\) is \(g(x) = 4x\). [3]
2. Using \(x_{n+1} = \ln 4x_n\), with \(x_0 = 2.15\), calculate \(x_1\), \(x_2\) and \(x_3\). Give the value of \(x_3\) to 4 decimal places. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$ Given that \(f(k) = 0\),

1. show, by calculation, that \(3.1 < k < 3.2\). [2]
2. Find \(f'(x)\). [3]

The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).

1. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\). [5]

The curve \(C\) with equation \(y = p + qe^x\), where \(p\) and \(q\) are constants, passes through the point \((0, 2)\). At the point \(P(\ln 2, p + 2q)\) on \(C\), the gradient is \(5\).

1. Find the value of \(p\) and the value of \(q\). [5]

The normal to \(C\) at \(P\) crosses the \(x\)-axis at \(L\) and the \(y\)-axis at \(M\).

1. Show that the area of \(\triangle OLM\), where \(O\) is the origin, is approximately \(53.8\). [5]

The curve with equation \(y = \ln 3x\) crosses the \(x\)-axis at the point \(P (p, 0)\).

1. Sketch the graph of \(y = \ln 3x\), showing the exact value of \(p\). [2]

The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.

1. Show that \(x = q\) is a solution of the equation \(x^2 + \ln 3x = 0\). [4]
2. Show that the equation in part (b) can be rearranged in the form \(x = \frac{1}{3}e^{-x^2}\). [2]
3. Use the iteration formula \(x_{n+1} = \frac{1}{3}e^{-x_n^2}\), with \(x_0 = \frac{1}{4}\), to find \(x_1, x_2, x_3\) and \(x_4\). Hence write down, to 3 decimal places, an approximation for \(q\). [3]

The curve \(C\) has equation \(y = f(x)\), where $$f(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$ The point \(P\) is a stationary point on \(C\).

1. Calculate the \(x\)-coordinate of \(P\). [4]
2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]

The point \(Q\) on \(C\) has \(x\)-coordinate \(1\).

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

The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).

1. Show that the \(x\)-coordinate of \(R\)
   1. satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
   2. lies between \(0.13\) and \(0.14\). [4]

Given that \(y = \log_a x, x > 0\), where \(a\) is a positive constant,

1. 1. express \(x\) in terms of \(a\) and \(y\), [1]
   2. deduce that \(\ln x = y \ln a\). [1]
2. Show that \(\frac{dy}{dx} = \frac{1}{x \ln a}\). [2]

The curve \(C\) has equation \(y = \log_{10} x, x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate \(10\). Using the result in part (b),

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

The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).

1. Find the exact \(x\)-coordinate of \(B\). [2]

The curve \(C\) is defined by the equation $$8x^3 - 3y^2 + 2xy = 9$$ Find an equation of the normal to \(C\) at the point \((2, 5)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

The curve \(C\) has the equation $$\sin(\pi y) - y - x^2 y = -5, \quad x > 0$$

1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5] The point \(P\) with coordinates \((2, 1)\) lies on \(C\). The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the exact value of the \(x\)-coordinate of \(A\). [4]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3^x$$ The point \(P\) lies on \(C\) and has coordinates \((2, 9)\). The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).

1. Find the exact value of the \(x\) coordinate of \(Q\). [4]

The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360°\) about the \(x\)-axis.

1. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac{p}{q}\) where \(p\) and \(q\) are exact constants. [You may assume the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] [6]

A curve has equation $$x^3 - 2xy - 4x + y^3 - 51 = 0.$$ Find an equation of the normal to the curve at the point \((4, 3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [8]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The rectangular hyperbola \(H\) has equation \(xy = 36\) The point \(P(4, 9)\) lies on \(H\)

1. Show, using calculus, that the normal to \(H\) at \(P\) has equation $$4x - 9y + 65 = 0$$ [4]

The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)

1. Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are rational constants. [5]

The point \(P(ap^2, 2ap)\) lies on the parabola \(M\) with equation \(y^2 = 4ax\), where \(a\) is a positive constant.

1. Show that an equation of the tangent to \(M\) at \(P\) is \(py = x + ap^2\). [3]

The point \(Q(16ap^2, 8ap)\) also lies on \(M\).

1. Write down an equation of the tangent to \(M\) at \(Q\). [2]

The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a constant.

1. Show that an equation for the normal to \(C\) at the point \(P(ap^2, 2ap)\) is \(y + px = 2ap + ap^3\). [4]

The normals to \(C\) at the points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), meet at the point \(R\).

1. Find, in terms of \(a\), \(p\) and \(q\), the coordinates of \(R\). [5]

The points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), lie on the parabola \(C\) with equation \(y^2 = 4ax\), where \(a\) is a constant.

1. Show that an equation for the chord \(PQ\) is \((p + q) y = 2(x + apq)\) . [3]

The normals to \(C\) at \(P\) and \(Q\) meet at the point \(R\).

1. Show that the coordinates of \(R\) are \((a(p^2 + q^2 + pq + 2), -apq(p + q) )\). [7]

The curve \(C\) has equation $$y = \operatorname{arcsec} e^x, \quad x > 0, \quad 0 < y < \frac{1}{2}\pi.$$

1. Prove that \(\frac{dy}{dx} = \frac{1}{\sqrt{e^{2x} - 1}}\). [5]
2. Sketch the graph of \(C\). [2]

The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).

1. Find the exact value of the \(y\)-coordinate of \(B\). [4]

The curve \(C\) has equation \(y = x^3 - 2x^2 - x + 3\) The point \(P\), which lies on \(C\), has coordinates \((2, 1)\).

1. Show that an equation of the tangent to \(C\) at the point \(P\) is \(y = 3x - 5\) [5]

The point \(Q\) also lies on \(C\). Given that the tangent to \(C\) at \(Q\) is parallel to the tangent to \(C\) at \(P\),

1. find the coordinates of the point \(Q\). [5]

The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),

1. find \(y\) in terms of \(x\). [4]
2. Verify that \(C\) passes through the point \((4, 0)\). [2]
3. Find an equation of the tangent to \(C\) at \(P\). [3]

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

1. Calculate the exact \(x\)-coordinate of \(Q\). [5]

A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).

1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]

The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.

1. Find the \(x\)-coordinate of \(Q\). [2]
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]

This tangent intersects the coordinate axes at the points \(R\) and \(S\).

1. Find the length of \(RS\), giving your answer as a surd. [4]

A circle with centre \(C\) has equation \(x^2 + y^2 - 2x + 10y - 19 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle. [3]
2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

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]

The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).

1. Find the value of the constant \(k\). [7]
2. Determine whether the stationary point is a maximum or minimum point. [2]
3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]

A curve has equation \(y = 3x^3 - 7x + \frac{2}{x}\).

1. Verify that the curve has a stationary point when \(x = 1\). [5]
2. Determine the nature of this stationary point. [2]
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]

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

1. Sketch the curve, giving the coordinates of all points of intersection with the axes. [3]
2. Find an equation of the tangent to the curve at the point where \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [9]