(a) 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 5 - 2006 - Paper 2

The missing values in the table can be calculated as follows:

• For $x = 0.2$: $3^{0.2} \approx 1.246$
• For $x = 0.4$: $3^{0.4} \approx 1.552$
• For $x = 0.6$: $3^{0.6} \approx 1.933$
• For $x = 0.8$: $3^{0.8} \approx 2.408$

The completed table is:

x	0	0.2	0.4	0.6	0.8	1
$3^x$	1	1.246	1.552	1.933	2.408	3

Using the trapezium rule, we approximate the integral as follows:

The formula for the trapezium rule for $n$ intervals is:

$\int_a^b f(x) \, dx \approx \frac{h}{2} (f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n))$

Here, $h = \frac{b-a}{n} = \frac{1-0}{5} = 0.2$ and the function values:

• $f(0) = 1$
• $f(0.2) = 1.246$
• $f(0.4) = 1.552$
• $f(0.6) = 1.933$
• $f(0.8) = 2.408$
• $f(1) = 3$

Now, applying the trapezium rule:

$\int_0^1 3^x \, dx \approx \frac{0.2}{2} (1 + 2(1.246 + 1.552 + 1.933 + 2.408) + 3)$

Calculating the above gives:

$\int_0^1 3^x \, dx \approx 0.1 (1 + 2(6.139) + 3) = 0.1 (1 + 12.278 + 3) = 0.1 (16.278) = 1.6278$

Thus, the approximate value is about $1.628$.