Sketch the curve with equation

$$y=k^x-7$$

where $k$ is a constant such that $k>1$.

On your sketch, show:

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

The table below shows the power output $P$ in kilowatts from a solar array, modeled by the equation $P(t)=3^t-1$, where $t$ is the time in hours after dawn.

Using the trapezium rule with all the values of $P$ in the given table, obtain an estimate for the total energy produced in the first two hours, given by

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

giving your answer to 2 decimal places.

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

(i)

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

(ii)

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