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 - Edexcel - A-Level Maths Pure - Question 7 - 2006 - Paper 2

$x$	$3^x$
0	1.000
0.2	1.245
0.4	1.515
0.6	1.933
0.8	2.408
1	3.000

The missing values were computed as follows:

• For $x = 0.2$, $3^{0.2} \approx 1.245$
• For $x = 0.4$, $3^{0.4} \approx 1.515$
• For $x = 0.6$, $3^{0.6} \approx 1.933$
• For $x = 0.8$, $3^{0.8} \approx 2.408$

To apply the trapezium rule:

1. We have 5 intervals with values $3^x$: 1.000, 1.245, 1.515, 1.933, 2.408, 3.000.

2. The trapezium rule formula:

   $ext{Area} \approx \frac{1}{2} \times h \times (f(a) + f(b))$

   where $h$ is the width of the sub-intervals. Here: $h = 0.2$.

Applying values: $\int_0^1 3^x \, dx \approx \frac{0.2}{2} \times (1 + 3) + (1.245 + 1.515 + 1.933 + 2.408)$ $= 0.1 \times (4 + 6.101) = 0.1 \times 10.101 = 1.0101$

After simplifying: $1.0101 \approx 1.8278$