1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)

4 The equation of a curve is \(y = \cos 2 x + 2 \sin x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the stationary points on the curve for \(0 < x < \pi\).

5 You are given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\).

1. Find \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = - 2 \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
3. Find a similar expression for \(\mathrm { f } ^ { \prime \prime \prime } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime \prime } ( 0 )\).
4. Find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).

3

1. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }\), find \(\mathrm { f } ^ { \prime } ( 0 )\) and \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
2. Hence find the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).

1 Given that $$y = x ^ { 2 } \sin x$$

1. show that the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) is \(\frac { 1 } { 2 } \pi\),
2. find the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

1. The height, \(h\) metres, of a plant, \(t\) years after it was first measured, is modelled by the equation

$$h = 2.3 - 1.7 \mathrm { e } ^ { - 0.2 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ Using the model,

1. find the height of the plant when it was first measured,
2. show that, exactly 4 years after it was first measured, the plant was growing at approximately 15.3 cm per year. According to the model, there is a limit to the height to which this plant can grow.
3. Deduce the value of this limit.

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
Given

• the point \(( 2,10 )\) lies on \(C\)
• the gradient of the curve at \(( 2,10 )\) is - 3
  1. (i) show that the value of \(a\) is - 2
     (ii) find the value of \(b\).
  2. Hence show that \(C\) has no stationary points.
  3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
  4. Hence deduce the coordinates of the points of intersection of the curve with equation

$$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\) c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\) Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x\) b. Find the coordinates of the points of inflection of the curve.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\) a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.

3. $$\mathrm { f } ( x ) = x + \tan \left( \frac { 1 } { 2 } x \right) \quad \pi < x < \frac { 3 \pi } { 2 }$$ Given that the equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\)

1. show that \(\alpha\) lies in the interval [3.6, 3.7]
2. Find \(\mathrm { f } ^ { \prime } ( x )\)
3. Using 3.7 as a first approximation for \(\alpha\), apply the Newton-Raphson method once to obtain a second approximation for \(\alpha\). Give your answer to 3 decimal places.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
      Given that \(l\) is a tangent to \(C\),
2. find the possible values of \(k\), giving your answers as simplified surds.

5. The mass, \(m\) grams, of a radioactive substance, \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { - 0.05 t }$$ According to the model,

1. find the mass of the radioactive substance six months after it was first observed,
2. show that \(\frac { \mathrm { d } m } { \mathrm {~d} t } = k m\), where \(k\) is a constant to be found.

11 A car is travelling along a stretch of road at a steady speed of \(11 \mathrm {~ms} ^ { - 1 }\).
The driver accelerates, and \(t\) seconds after starting to accelerate the speed of the car, \(V\), is modelled by the formula \(\mathrm { V } = \mathrm { A } + \mathrm { B } \left( 1 - \mathrm { e } ^ { - 0.17 \mathrm { t } } \right)\).
When \(t = 3 , V = 13.8\).

1. Find the values of \(A\) and \(B\), giving your answers correct to 2 significant figures. When \(t = 4 , V = 14.5\) and when \(t = 5 , V = 14.9\).
2. Determine whether the model is a good fit for these data.
3. Determine the acceleration of the car according to the model when \(t = 5\), giving your answer correct to 3 decimal places. The car continues to accelerate until it reaches its maximum speed.
   The speed limit on this road is \(60 \mathrm { kmh } ^ { - 1 }\). All drivers who exceed this speed limit are recorded by a speed camera and automatically fined \(\pounds 100\).
4. Determine whether, according to the model, the driver of this car is fined \(\pounds 100\).

14 Alex places a hot object into iced water and records the temperature \(\theta ^ { \circ } \mathrm { C }\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta _ { 0 } \mathrm { e } ^ { - k t }\) where \(\theta _ { 0 }\) and \(k\) are constants whose values depend on the characteristics of the object. The temperature of Alex's object is \(82 ^ { \circ } \mathrm { C }\) when it is placed into the water. After 5 minutes the temperature is \(27 ^ { \circ } \mathrm { C }\).

1. Find the values of \(\theta _ { 0 }\) and \(k\) that best model the data.
2. Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water.
3. Show that the model with the values found in part (a) can be written as \(\ln \theta = \mathrm { a } -\) bt where \(a\) and \(b\) are constants to be determined. Ben places a different object into iced water at the same time as Alex. The model for Ben's object is \(\ln \theta = 3.4 - 0.08 t\).
4. Determine each of the following:
   Find this time and the corresponding temperature.

10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.

1. 1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
   2. Give the equations of any asymptotes.
2. What does the model predict about the long term population of
   Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
3. Comment on the validity of the model given by the equations, giving a reason for your answer.
4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).

3 You are given that \(y = 4 x + \sin 8 x\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the smallest positive value of \(x\) for which \(\frac { \mathrm { dy } } { \mathrm { dx } } = 0\), giving your answer in an exact form.

9 Show that \(\mathrm { y } = \mathrm { x }\) has the same gradient as \(\mathrm { y } = \sin \mathrm { x }\) when \(\mathrm { x } = 0\), as stated in line 5 .

11 Fig. 11.1 shows the curve with equation \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) where \(\mathrm { g } ( x ) = x \sin x + \cos x\) and the curve of the gradient function \(\mathrm { y } = \mathrm { g } ^ { \prime } ( \mathrm { x } )\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\). \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinates of the points on the curve \(y = g ( x )\) where the gradient is 1 satisfy the equation \(\frac { 1 } { x } - \cos x = 0\). Fig. 11.2 shows part of the curve with equation \(y = \frac { 1 } { x } - \cos x\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 11.2} \includegraphics[alt={},max width=\textwidth]{60e1e785-c34b-48ef-a63f-13a25fee186e-09_678_1363_424_239}
   \end{figure}
2. Use the Newton-Raphson method with a suitable starting value to find the smallest positive \(x\)-coordinate of a point on the curve \(y = x \sin x + \cos x\) where the gradient is 1 . You should write down at least the following.

11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).

1. 1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
   2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
2. Would the result still hold with a larger domain? Give reasons for your answer.

9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}

1. 1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
   2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
2. 1. Write down the \(y\)-coordinate of \(A\).
   2. Show that \(x = \ln 2\) at \(B\).
3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.

3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(\quad y = \ln ( 5 x - 2 )\);
   2. \(y = \sin 2 x\).
2. The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \ln ( 5 x - 2 ) , & \text { for real values of } x \text { such that } x \geqslant \frac { 1 } { 2 } \\ \mathrm {~g} ( x ) = \sin 2 x , & \text { for real values of } x \text { in the interval } - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } \end{array}$$
   1. Find the range of f .
   2. Find an expression for \(\operatorname { gf } ( x )\).
   3. Solve the equation \(\operatorname { gf } ( x ) = 0\).
   4. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

2

1. Given that \(y = x ^ { 4 } \tan 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. Find the gradient of the curve with equation \(y = \frac { x ^ { 2 } } { x - 1 }\) at the point where \(x = 3\).
   (3 marks)

1. (a) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that

$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(b) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
4. Determine whether the stationary point is a maximum or minimum point and justify your answer.