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 6 - 2006 - Paper 2

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

• For $x = 0.2$, $3^{0.2} \approx 1.245729$ (rounded to $1.246$).
• For $x = 0.4$, $3^{0.4} \approx 1.515716$ (rounded to $1.516$).
• For $x = 0.6$, $3^{0.6} \approx 1.933$.
• For $x = 0.8$, $3^{0.8} \approx 2.408$.

The completed table is:

$x$	$3^x$
0	1.000
0.2	1.246
0.4	1.516
0.6	1.933
0.8	2.408
1	3.000

To approximate the integral using the trapezium rule, we can use the formula:

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

In this case, with $a = 0$, $b = 1$, and $n = 5$:

• $f(0) = 3^0 = 1$
• $f(0.2) = 1.246$
• $f(0.4) = 1.516$
• $f(0.6) = 1.933$
• $f(0.8) = 2.408$
• $f(1) = 3$

Now applying the trapezium rule:

$= \frac{1-0}{2} (1 + 3) + (1 - 0) \left(1.246 + 1.516 + 1.933 + 2.408\right)$

Calculating gives: $= 0.5 \times 4 + (1.246 + 1.516 + 1.933 + 2.408)$ $= 2 + 7.103 = 9.103$

Now we divide by 2 (since segments are halved by the trapezium rule): $\frac{9.103}{2} \approx 4.5515$.

Thus, the final approximation is about $1.8278$.