1.07l Derivative of ln(x): and related functions

9

1. Given that \(y = \frac { 4 x } { 4 x - 3 }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( 4 x - 3 ) ^ { 2 } }\), where \(k\) is an integer.
2. 1. Given that \(y = x \ln ( 4 x - 3 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln ( 4 x - 3 )\) at the point where \(x = 1\).
3. 1. Use the substitution \(u = 4 x - 3\) to find \(\int \frac { 4 x } { 4 x - 3 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   2. By using integration by parts, or otherwise, find \(\int \ln ( 4 x - 3 ) \mathrm { d } x\).

1

1. Differentiate \(\ln x\) with respect to \(x\).
2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).

10 A curve has equation \(x = ( y + 5 ) \ln ( 2 y - 7 )\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of y .
2. Find the gradient of the curve where it crosses the y -axis.

2 Given that \(y = \ln ( 5 x )\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

7

1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).

8 Show that the graph of \(y = x ^ { 2 } - \ln x\) has only one stationary point and give the coordinates of that point in exact form.

9

1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).

12

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
2. Find
   1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
   2. \(\int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).

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 $$2 y - x = \ln ( 16 ) - 1$$

10

1. 1. By writing \(\sec x = \frac { 1 } { \cos x }\), prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x .$$
   2. Deduce that \(y = \sec x\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \sqrt { y ^ { 2 } - 1 } , \quad 0 \leq x < \frac { 1 } { 2 } \pi .$$
2. A curve lies in the first quadrant of the cartesian plane with origin \(O\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-4_707_698_845_685} The normal to the curve at the point \(P ( x , y )\) meets the \(x\)-axis at the point \(Q\). The angle between \(O P\) and the \(x\)-axis is \(u\), and the angle between \(Q P\) and the \(x\)-axis is \(v\).
   1. If $$\tan v = \tan ^ { 2 } u$$ obtain a differential equation satisfied by the curve.
   2. The curve passes through the point \(( 2,1 )\). By solving the differential equation, find an equation for the curve in the implicit form $$\mathrm { F } ( x , y ) = C ,$$ where \(C\) is a constant that should be determined.

A curve has equation \(y = \frac{4}{3x - 4}\) and \(P(2, 2)\) is a point on the curve.

1. Find the equation of the tangent to the curve at \(P\). [4]
2. Find the angle that this tangent makes with the \(x\)-axis. [2]

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]

The equation of a curve is \(y = \frac{A}{2x - 1}\).

1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360°\) about the \(x\)-axis. [4]
2. Given that the line \(2y = x + c\) is a normal to the curve, find the possible values of the constant \(c\). [6]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
2. Find the coordinates of \(M\). [3]
3. Find, showing all necessary working, the area of the shaded region. [4]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
4. Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]

A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Find the exact value of \(t\) at the stationary point. [2]
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).

1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]

\includegraphics{figure_8} The diagram shows a sketch of the curve \(y = x^2\ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \((1, 0)\).

1. Find the exact value of the \(x\)-coordinate of \(M\). [4]
2. Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places. [5]

The curve \(y = \frac{\ln x}{x + 1}\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac{x + 1}{\ln x},$$ and that this \(x\)-coordinate lies between 3 and 4. [5]
2. Use the iterative formula $$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]