A scientist is studying the potential of a chemical reaction, modeled by the equation

$$V=k^t+12$$

where $k$ is a constant such that $k>1$ and $t$ is time. Sketch the graph of $V$ against $t$.

On your sketch, show:

• the coordinates of the point of intersection of the curve with the $V$-axis
• the equation of the horizontal asymptote of the curve.

The table shows corresponding values of time $t$ and the rate of airflow $R(t)$ in a ventilation shaft, where

$$R(t)=3^t-4$$

Using the trapezium rule with all the values of $R(t)$ in the given table, obtain an estimate for

$${\int }_0^2(3^t-4)dt$$

giving your answer to 2 decimal places.

Using your answer to part (b) and making your method clear, estimate

(i)

$${\int }_0^2(3^t+2)dt$$

(ii)

$${\int }_0^2(3^{t+1}-12)dt$$