1.07j Differentiate exponentials: e^(kx) and a^(kx)

5 \includegraphics[max width=\textwidth, alt={}, center]{15dd10f9-73d4-4107-bb45-7866f5470572-3_606_890_815_630} The diagram shows the curve with equation \(y = x \mathrm { e } ^ { - x } + 1\). The curve crosses the \(x\)-axis at \(x = \alpha\).

1. Use differentiation to show that the \(x\)-coordinate of the stationary point is 1 . \(\alpha\) is to be found using the Newton-Raphson method, with \(\mathrm { f } ( x ) = x \mathrm { e } ^ { - x } + 1\).
2. Explain why this method will not converge to \(\alpha\) if an initial approximation \(x _ { 1 }\) is chosen such that \(x _ { 1 } > 1\).
3. Use this method, with a first approximation \(x _ { 1 } = 0\), to find the next three approximations \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Find \(\alpha\), correct to 3 decimal places.

4.A curve \(C\) has equation \(y = \mathrm { f } ( x )\) with \(\mathrm { f } ^ { \prime } ( x ) > 0\) .The \(x\)-coordinate of the point \(P\) on the curve is \(a\) .The tangent and the normal to \(C\) are drawn at \(P\) .The tangent cuts the \(x\)-axis at the point \(A\) and the normal cuts the \(x\)-axis at the point \(B\) .

1. Show that the area of \(\triangle A P B\) is $$\frac { 1 } { 2 } [ \mathrm { f } ( a ) ] ^ { 2 } \left( \frac { \left[ \mathrm { f } ^ { \prime } ( a ) \right] ^ { 2 } + 1 } { \mathrm { f } ^ { \prime } ( a ) } \right)$$
2. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 5 x }\) and the area of \(\triangle A P B\) is \(\mathrm { e } ^ { 5 a }\) ,find and simplify the exact value of \(a\) .

5 The mass, \(M\) grams, of a certain substance is increasing exponentially so that, at time \(t\) hours, the mass is given by $$M = 40 \mathrm { e } ^ { k t }$$ where \(k\) is a constant. The following table shows certain values of \(t\) and \(M\).

1. In either order,
   1. find the values missing from the table,
   2. determine the value of \(k\).
   3. Find the rate at which the mass is increasing when \(t = 21\).

9

1. The function f is defined for all real values of \(x\) by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$
   1. Show that \(\mathrm { f } ^ { \prime } ( x ) > 0\) for all \(x\).
   2. Show that the set of values of \(x\) for which \(\mathrm { f } ^ { \prime \prime } ( x ) > 0\) is the same as the set of values of \(x\) for which \(\mathrm { f } ( x ) > 0\), and state what this set of values is.
   3. \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where \(k\) is a constant greater than 1 . The graph of \(y = \mathrm { g } ( x )\) is shown above. Find the range of g , giving your answer in simplified form.

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1. Substance \(A\) is decaying exponentially and its mass is recorded at regular intervals. At time \(t\) years, the mass, \(M\) grams, of substance \(A\) is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$
   1. Find the time taken for the mass of substance \(A\) to decrease to \(25 \%\) of its value when \(t = 0\).
   2. Find the rate at which the mass of substance \(A\) is decreasing when \(t = 5\).
   3. Substance \(B\) is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance \(B\) after a further year.

9 Fig. 9 shows the curve \(y = \frac { x ^ { 2 } } { 3 x - 1 }\).
P is a turning point, and the curve has a vertical asymptote \(x = a\). \begin{figure}[h] \end{figure}

1. Write down the value of \(a\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ( 3 x - 2 ) } { ( 3 x - 1 ) ^ { 2 } }\).
3. Find the exact coordinates of the turning point P . Calculate the gradient of the curve when \(x = 0.6\) and \(x = 0.8\), and hence verify that P is a minimum point.
4. Using the substitution \(u = 3 x - 1\), show that \(\int \frac { x ^ { 2 } } { 3 x - 1 } \mathrm {~d} x = \frac { 1 } { 27 } \int \left( u + 2 + \frac { 1 } { u } \right) \mathrm { d } u\). Hence find the exact area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = \frac { 2 } { 3 }\) and \(x = 1\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h] \end{figure}

1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
2. Find the exact coordinates of the turning point P .
3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( e ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\)-and \(y\)-axes.

1 Fig. 1 shows part of the curve \(y = \mathrm { e } ^ { 2 x } \cos x\). \begin{figure}[h] \end{figure} Find the coordinates of the turning point P .

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1. State the derivative of \(\mathrm { e } ^ { \cos x }\).
2. Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

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 \(\mathrm { f } ( x )\).

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points of \(C\).
3. Draw a sketch of \(C\).

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1. Differentiate \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) with respect to \(x\).
2. Hence show that \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) is increasing for all values of \(x\).

1 Differentiate the following with respect to \(x\).

1. \(\mathrm { e } ^ { - 4 x }\)
2. \(\frac { x ^ { 2 } } { x + 1 }\)

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation $$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$ where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)

1. state the value of \(k\). From the instant when the tyre developed the puncture,
2. find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\) Give your answer in minutes to one decimal place.
3. Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
   Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

14. A population of ants being studied on an island. The number of ants, \(P\), in the population, is modelled by the equation. $$P = \frac { 900 k e ^ { 0.2 t } } { 1 + k e ^ { 0.2 t } } , \text { where } k \text { is a constant. }$$ Given that there were 360 ants when the study started,
a. show that \(k = \frac { 2 } { 3 }\).
b. Show that \(P = \frac { 1800 } { 2 + 3 e ^ { - 0.2 t } }\). The model predicts an upper limit to the number of ants on the island.
c. State the value of this limit.
d. Find the value of \(t\) when \(P = 520\). Give your answer to one decimal place.
e. i. Show that the rate of growth, \(\frac { \mathrm { d } P } { d t } = \frac { P ( 900 - P ) } { 4500 }\) ii. Hence state the value of \(P\) at which the rate of growth is a maximum.

12. A curve has equation \(y = \frac { 2 x e ^ { x } } { x + k }\) where \(k\) is a positive constant.
i. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { e ^ { x } \left( 2 x ^ { 2 } + 2 k x + 2 k \right) } { ( x + k ) ^ { 2 } }\) ii. Given that the curve has exactly one stationary point find the value of \(k\).

1. The function f is defined by

$$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$ where \(k\) is a positive constant.

1. Show that $$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$ where \(\mathrm { g } ( x )\) is a function to be found. Given that the curve with equation \(y = \mathrm { f } ( x )\) has at least one stationary point, (b) find the range of possible values of \(k\).

11. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = 8 x ^ { 2 } \mathrm { e } ^ { - 3 x } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 5, is bounded by

• the curve \(C\)
• the line with equation \(x = 1\)
• the \(x\)-axis

Find the exact area of \(R\), giving your answer in the form $$A + B \mathrm { e } ^ { - 3 }$$ where \(A\) and \(B\) are rational numbers to be found.

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1. Write down an expression for the gradient of the curve \(y = \mathrm { e } ^ { k x }\).
2. The line L is a tangent to the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) at the point where \(x = 2\). Show that L passes through the point \(( 0,0 )\).
3. Find the coordinates of the point of intersection of the curves \(y = 3 \mathrm { e } ^ { x }\) and \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.

1. Write down the value of \(a\).
2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
3. Calculate the value of \(k\).
4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.

8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes. Fig. 8 shows the experimental data. \begin{figure}[h] \end{figure} It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.

1. Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
2. Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
3. Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
   1. \(\quad t = 0\),
   2. \(t = 20\).
4. Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.

9 The gradient of a curve is given by \(\frac { d y } { d x } = e ^ { x } - 4 e ^ { - x }\).

1. Show that the \(x\)-coordinate of any point on the curve at which the gradient is 3 satisfies the equation \(\left( e ^ { x } \right) ^ { 2 } - 3 e ^ { x } - 4 = 0\).
2. Hence show that there is only one point on the curve at which the gradient is 3 , stating the exact value of its \(x\)-coordinate.
3. The curve passes through the point \(( 0,0 )\). Show that when \(x = 1\) the curve is below the \(x\)-axis.