The total number of bacteria at a colony of the day is shown in the table below.

$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline t \text{ (times)} & 10 am & 11 am & 12 pm & 1 pm & 2 pm & 3 pm & 4 pm \\ \hline
N \text{ (quantity) } & 10 & 70 & 120 & 190 & 230 & 170 & 140 \\ \hline \end{array} $$

Find \(A \) and \(B \) if the average rate of change between \(x = 2 \) and \(x = 2 + h\) for \(y = 2{x^2} – 4x + 1\) is \(A + Bh\).

Hence, find the gradient of the tangent to \(y = 2{x^2} – 4x + 1\) at \(x = 2 \).

Assume an oil spill from an oil tanker is circular and remains that way. Find the rate of change of \(A\), the area of the spill, when the radius is 10, correcting to three significant figures.

A particle  \(P\)   moves in a straight line with a velocity function of  \(s\left( t \right) = {t^3} – 3{t^2} – 9t + 4\) metres,  where  \(t \ge 0\),  \(t\)   in seconds.    Find the total distance travelled for the first 5 seconds.

A particle  \(P\)   moves in a straight line with a velocity function of  \(s\left( t \right) = {t^2} – 4t – 5\) metres,  where  \(t \ge 0\),  \(t\)   in seconds.    Find the total distance travelled in the time from  \(t = 1\)  and  \(t = 4\).

The displacement of a particle \(x\left( t \right)\) metres in a straight line from a fixed point \( O\), the origin, at any time \(t \) seconds is given by \(x\left( t \right) = {t^2} – 8t + 2\). Find the total distance traveled between \(t = 1\) and \(t = 2\).

The amount in an investment account is initially \($5000\) and increase by 7% per annum. Find the amount in the account after 6 years, correcting to the nearest dollar.

The amount in an investment account is initially \( $4500 \) at the beginning of January 2019 and increase by 5% per annum. At which year and month the amount becomes double.

The value of a scrap metal bought for \($3400 \) decreases by 3% each year. Find how much the metal is worth after 7 years, correcting to 2 significant figures.

The mass of radioactive \(t \) days after establishment is \({R_t} = 500 \times {0.7^{2.5t}}\) grams. Find the time for the mass to reach 200 grams. Answer correct to the nearest hour.