Sketch exponential graphs

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 \(\int _ { 0 } ^ { 1 } 3 ^ { x } \mathrm {~d} x\).

6

1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
3. Sketch the curve with equation \(y = 2 ^ { x }\).

6

1. On the same axes, sketch the curves \(y = 3 ^ { x }\) and \(y = 3 ^ { 2 x }\), identifying clearly which is which.
2. Given that \(3 ^ { 2 x } = 729\), find in either order the values of \(3 ^ { x }\) and \(x\).

1. (a) Sketch the curve with equation

$$y = 4 ^ { x }$$ stating any points of intersection with the coordinate axes.
(b) Solve $$4 ^ { x } = 100$$ giving your answer to 2 decimal places.

8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).

1. Find the value of the \(y\)-coordinate of \(A\).
2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1 \\ - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
5. 1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
   2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).

2

1. Sketch the graph of \(y = ( 0.2 ) ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(( 0.2 ) ^ { x } = 4\), giving your answer to three significant figures.
3. Describe the geometrical transformation that maps the graph of \(y = ( 0.2 ) ^ { x }\) onto the graph of \(y = 5 ^ { x }\).
   [0pt] [1 mark]

11. (a) Sketch the graph of \(y = 3 ^ { x }\). Clearly label the coordinates of the point where the graph crosses the \(y\)-axis.
(b) On the same set of axes, sketch the graph of \(y = 3 ^ { ( x + 1 ) }\), clearly labelling the coordinates of the point where the graph crosses the \(y\)-axis.

2 The graph of \(y = 5 ^ { x }\) is transformed by a stretch in the \(y\)-direction, scale factor 5 State the equation of the transformed graph. Circle your answer.
[0pt] [1 mark] \(y = 5 \times 5 ^ { x }\) \(y = 5 ^ { \frac { x } { 5 } }\) \(y = \frac { 1 } { 5 } \times 5 ^ { x }\) \(y = 5 ^ { 5 x }\)

14

1. 1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
2. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
3. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
4. 1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\) 14
5. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\) Find the exact value of the total area of the eight rectangles.
   Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
   [0pt] [3 marks]
   14
6. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)

1 Which one of these functions is decreasing for all real values of \(x\) ?
Circle your answer.
[0pt] [1 mark] $$\mathrm { f } ( x ) = \mathrm { e } ^ { x } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { 1 - x } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { x - 1 } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { - x }$$