1.06a Exponential function: a^x and e^x graphs and properties

1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).

1 Sketch the graph of \(y = \mathrm { e } ^ { a x } - 1\) where \(a\) is a positive constant.

7 \includegraphics[max width=\textwidth, alt={}, center]{25dffd43-9456-449b-be77-8402109ee603-3_608_672_283_733} The diagram shows the curve \(y = 2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - 2 x }\). The curve cuts the \(y\)-axis at \(A\).

1. Write down the coordinates of \(A\).
2. Find the equation of the tangent to the curve at \(A\), and state the coordinates of the point where this tangent meets the \(x\)-axis.
3. Calculate the area of the region bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = 1\), giving your answer correct to 2 significant figures.

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 $$\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \frac { 1 } { 1 + \cos 4 x } \mathrm {~d} x$$

1 Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$

5

1. By sketching a suitable pair of graphs, show that the equation \(\operatorname { cosec } x = 1 + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) has exactly two roots in the interval \(0 < x < \pi\).
2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \pi - \sin ^ { - 1 } \left( \frac { 1 } { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } + 1 } \right)$$ with initial value \(x _ { 1 } = 2\), converges to one of these roots.
   Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

8 In a certain chemical reaction the amount, \(x\) grams, of a substance is increasing. The differential equation satisfied by \(x\) and \(t\), the time in seconds since the reaction began, is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k x \mathrm { e } ^ { - 0.1 t }$$ where \(k\) is a positive constant. It is given that \(x = 20\) at the start of the reaction.

1. Solve the differential equation, obtaining a relation between \(x , t\) and \(k\).
2. Given that \(x = 40\) when \(t = 10\), find the value of \(k\) and find the value approached by \(x\) as \(t\) becomes large.

8 The curve with equation \(y = \frac { x ^ { 3 } } { \mathrm { e } ^ { x } - 1 }\) has a stationary point at \(x = p\), where \(p > 0\).

1. Show that \(p = 3 \left( 1 - \mathrm { e } ^ { - p } \right)\).
2. Verify by calculation that \(p\) lies between 2.5 and 3 .
3. Use an iterative formula based on the equation in part (a) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).

2. Given \(y = 3 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.

1. \(3 ^ { 3 x }\)
2. \(\frac { 1 } { 3 ^ { x - 2 } }\)
3. \(\frac { 81 } { 9 ^ { 2 - 3 x } }\)

2. Given \(y = 2 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.

1. \(2 ^ { 2 x }\)
2. \(2 ^ { x + 3 }\)
3. \(\frac { 1 } { 4 ^ { 2 x - 3 } }\)

1. (a) Sketch the graph of \(y = 3 ^ { x - 2 } , x \in \mathbb { R }\)

Give the exact values for the coordinates of the point where your graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = 3 ^ { x - 2 }\) The values of \(y\) are rounded to 3 decimal places where necessary. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { 0.5 } ^ { 3 } 3 ^ { x - 2 } \mathrm {~d} x$$ Give your answer to 2 decimal places.

1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of

$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\) (b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for

1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
   \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}

6. (a) Sketch the graph of \(y = \left( \frac { 1 } { 2 } \right) ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are rounded to 3 decimal places. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { - 0.9 } ^ { - 0.5 } \left( \frac { 1 } { 2 } \right) ^ { x } d x$$ II

9. (a) Sketch the curve with equation $$y = 3 \times 4 ^ { x }$$ showing the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 6 ^ { 1 - x }\) meets the curve with equation \(y = 3 \times 4 ^ { x }\) at the point \(P\).
(b) Show that the \(x\) coordinate of \(P\) is \(\frac { \log _ { 10 } 2 } { \log _ { 10 } 24 }\)

1. (a) Sketch the curve with equation

$$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\) On your sketch show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of the asymptote to the curve.

The table above shows corresponding values of \(x\) and \(y\) for \(y = 3 ^ { - \frac { 1 } { 2 } x } + 4\) The values of \(y\) are given to four significant figures, as appropriate.
Using the trapezium rule with all the values of \(y\) in the table,
(b) find an approximate value for $$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(c) Using the answer to part (b), find an approximate value for

1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 8.5 } ^ { 4 } \left( 3 ^ { \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.

1. On Diagram 1, sketch the curve with equation
   1. \(y = 2 ^ { x }\)
   2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
2. Using algebra, find the exact coordinates of \(P\).
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
   \section*{Diagram 1}

1. (a) Sketch the curve with equation

$$y = a ^ { x } + 4$$ where \(a\) is a positive constant greater than 1
On your sketch, show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of the asymptote of the curve

The table shows corresponding values of \(x\) and \(y\) for $$y = 2 ^ { x } - 2 x$$ with the values of \(y\) given to 4 decimal places as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } - 2 x \right) \mathrm { d } x\), giving your answer to 2 decimal places.
(c) Using your answer to part (b) and making your method clear, estimate

1. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x } + 2 x \right) \mathrm { d } x\)
2. \(\int _ { 2 } ^ { 3.5 } \left( 2 ^ { x + 1 } - 4 x \right) \mathrm { d } x\)

5. (a) Given \(0 < a < 1\), sketch the curve with equation $$y = a ^ { x }$$ showing the coordinates of the point at which the curve crosses the \(y\)-axis. The table above shows corresponding values of \(x\) and \(y\) for \(y = x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x }\) The values of \(y\) are given to 4 significant figures as appropriate.
Using the trapezium rule with all the values of \(y\) in the given table,
(b) obtain an estimate for \(\int _ { 2 } ^ { 4 } \left( x ^ { 2 } + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\) Using your answer to part (b) and making your method clear, estimate
(c) \(\quad \int _ { 2 } ^ { 4 } \left( x ( x - 3 ) + \left( \frac { 1 } { 2 } \right) ^ { x } \right) \mathrm { d } x\)

1. (a) Sketch the graph of \(y = 7 ^ { x } , x \in \mathbb { R }\), showing the coordinates of any points at which the graph crosses the axes.
   (b) Solve the equation

$$7 ^ { 2 x } - 4 \left( 7 ^ { x } \right) + 3 = 0$$ giving your answers to 2 decimal places where appropriate.

5. (a) In the space provided, sketch the graph of \(y = 3 ^ { x } , x \in \mathbb { R }\), showing the coordinates of the point at which the graph meets the \(y\)-axis.
(b) Complete the table, giving the values of \(3 ^ { x }\) to 3 decimal places. (c) Use the trapezium rule, with all the values from your table, to find an approximation for the value of \(\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x\).

4. (a) Complete the table below, giving values of \(\sqrt { } \left( 2 ^ { x } + 1 \right)\) to 3 decimal places. \begin{figure}[h] \end{figure} Figure 1 shows the region \(R\) which is bounded by the curve with equation \(y = \sqrt { } \left( 2 ^ { x } + 1 \right)\), the \(x\)-axis and the lines \(x = 0\) and \(x = 3\) (b) Use the trapezium rule, with all the values from your table, to find an approximation for the area of \(R\).
(c) By reference to the curve in Figure 1 state, giving a reason, whether your approximation in part (b) is an overestimate or an underestimate for the area of \(R\).

8. (a) Sketch the graph of $$y = 3 ^ { x } , \quad x \in \mathbb { R }$$ showing the coordinates of any points at which the graph crosses the axes.
(b) Use algebra to solve the equation $$3 ^ { 2 x } - 9 \left( 3 ^ { x } \right) + 18 = 0$$ giving your answers to 2 decimal places where appropriate.

6. \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 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 e ^ { 3 \sin x } \cos x \quad 0 \leqslant x \leqslant 2 \pi$$ The curve intersects the \(x\)-axis at point \(R\), as shown in Figure 1.

1. State the coordinates of \(R\) The curve has two turning points, at point \(P\) and point \(Q\), also shown in Figure 1.
2. Show that, at points \(P\) and \(Q\), $$a \sin ^ { 2 } x + b \sin x + c = 0$$ where \(a\), \(b\) and \(c\) are integers to be found.
3. Hence find the \(x\) coordinate of point \(Q\), giving your answer to 3 decimal places.