1.07q Product and quotient rules: differentiation

2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \ln ( 1 + \sin 2 x )\),
2. \(y = \frac { \tan x } { x }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-3_451_933_1777_605} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\) is equal to \(2 - \frac { 17 } { \mathrm { e } ^ { 3 } }\).
2. Find the \(x\)-coordinate of the maximum point \(M\) on the curve.
3. Find the \(x\)-coordinate of the point \(P\) at which the tangent to the curve passes through the origin.

2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.

4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { x ^ { 3 } }\) has one stationary point.

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

5 For each of the following curves, find the gradient at the point where the curve crosses the \(y\)-axis:

1. \(y = \frac { 1 + x ^ { 2 } } { 1 + \mathrm { e } ^ { 2 x } }\);
2. \(2 x ^ { 3 } + 5 x y + y ^ { 3 } = 8\).

10 \includegraphics[max width=\textwidth, alt={}, center]{b6bede75-3da4-4dda-9303-a5a692fc2572-3_556_1093_1596_523} The diagram shows the curve \(y = 10 e ^ { - \frac { 1 } { 2 } x } \sin 4 x\) for \(x \geqslant 0\). The stationary points are labelled \(T _ { 1 } , T _ { 2 }\), \(T _ { 3 } , \ldots\) as shown.

1. Find the \(x\)-coordinates of \(T _ { 1 }\) and \(T _ { 2 }\), giving each \(x\)-coordinate correct to 3 decimal places.
2. It is given that the \(x\)-coordinate of \(T _ { n }\) is greater than 25 . Find the least possible value of \(n\).

4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).

3 The curve \(y = \frac { \mathrm { e } ^ { x } } { \cos x }\), for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\), has one stationary point. Find the \(x\)-coordinate of this point.

4 A curve has equation \(y = \mathrm { e } ^ { - 3 x } \tan x\). Find the \(x\)-coordinates of the stationary points on the curve in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\). Give your answers correct to 3 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{7fe27759-d014-4bc6-8391-342d9df8280e-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{346e8866-ca23-4ea6-81bf-bf62502a16d1-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

5 The expression \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x \mathrm { e } ^ { - 2 x }\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( - \frac { 1 } { 2 } \right)\).
2. Find the exact value of \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x\).

5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. Explain why the gradient of the curve is never negative.
3. Find the value of \(x\) for which the gradient is least.

3 A curve has equation $$y = \frac { 2 - \tan x } { 1 + \tan x }$$ Find the equation of the tangent to the curve at the point for which \(x = \frac { 1 } { 4 } \pi\), giving the answer in the form \(y = m x + c\) where \(c\) is correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

7 \includegraphics[max width=\textwidth, alt={}, center]{84df6b9a-6118-44a2-9c18-512039ded4fd-3_416_677_258_733} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

2 The equation of a curve is \(y = \frac { \sin x } { 1 + \cos x }\), for \(- \pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.

2 The curve with equation \(y = \frac { \mathrm { e } ^ { - 2 x } } { 1 - x ^ { 2 } }\) has a stationary point in the interval \(- 1 < x < 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of this stationary point, giving the answer correct to 3 decimal places.

5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. Explain why the gradient of the curve is never negative.
3. Find the value of \(x\) for which the gradient is least.

2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } } { d x ^ { n } } \left( x ^ { 2 } e ^ { x } \right) = \left( x ^ { 2 } + 2 n x + n ( n - 1 ) \right) e ^ { x }$$

6 \includegraphics[max width=\textwidth, alt={}, center]{6694ccc1-c8b1-42a7-8b21-829a89af74c9-08_732_807_258_667} The diagram shows the curve with equation \(y = \frac { 8 + x ^ { 3 } } { 2 - 5 x }\). The maximum point is denoted by \(M\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the gradient of the curve at the point where the curve crosses the \(x\)-axis.
2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \sqrt { } \left( 0.6 x + 4 x ^ { - 1 } \right)\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{d53a2d6b-4c5e-4bc6-8aa1-587e97c87920-2_371_839_1409_651} The diagram shows the part of the curve \(y = 3 \mathrm { e } ^ { - x } \sin 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and the stationary point \(M\).

1. Find the equation of the tangent to the curve at the origin.
2. Find the coordinates of \(M\), giving each coordinate correct to 3 decimal places.

5 The curve with equation \(y = x ^ { 2 } \ln x\), where \(x > 0\), has one stationary point.

1. Find the \(x\)-coordinate of this point, giving your answer in terms of e .
2. Determine whether this point is a maximum or a minimum point.

6 \includegraphics[max width=\textwidth, alt={}, center]{4029c46c-50a1-4d23-bc29-589417a6b7f5-3_501_497_269_826} The diagram shows the part of the curve \(y = \frac { \mathrm { e } ^ { 2 x } } { x }\) for \(x > 0\), and its minimum point \(M\).

1. Find the coordinates of \(M\).
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 1 } ^ { 2 } \frac { \mathrm { e } ^ { 2 x } } { x } \mathrm {~d} x$$ giving your answer correct to 1 decimal place.
3. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).
4. Given that \(y = \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
5. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sec ^ { 2 } 2 x \mathrm {~d} x = \frac { 1 } { 2 } \sqrt { } 3$$ and, by using an appropriate trigonometrical identity, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan ^ { 2 } 2 x \mathrm {~d} x\).
6. Use the identity \(\cos 4 x \equiv 2 \cos ^ { 2 } 2 x - 1\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$

8 \includegraphics[max width=\textwidth, alt={}, center]{8f815127-61b2-4a7f-8687-747950ea6597-3_693_1061_262_541} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\) and its maximum point \(M\).

1. Find the \(x\)-coordinate of \(M\).
2. Show that the tangent to the curve at the point where \(x = 1\) passes through the origin.
3. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.