Air is pumped into a spherical balloon at the rate of \(250 \mathrm {~cm} ^ { 3 }\) per second. When the radius of the balloon is 15 cm , calculate the rate at which the radius is increasing, giving your answer to three decimal places
(a) Sketch the graph of \(y = x ^ { 2 } + 6 x + 13\), identifying the stationary point.
(b) The function \(f\) is defined by \(f ( x ) = x ^ { 2 } + 6 x + 13\) with domain \(( a , b )\).
Explain why \(f ^ { - 1 }\) does not exist when \(a = - 10\) and \(b = 10\).
Write down a value of \(a\) and a value of \(b\) for which the inverse of \(f\) does exist and derive an expression for \(f ^ { - 1 } ( x )\).
(a) Expand \(( 1 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending power of \(x\) as far as the term in \(x ^ { 2 }\). State the range of \(x\) for which the expansion is valid.
(b) By taking \(x = \frac { 1 } { 10 }\), find an approximation for \(\sqrt { 10 }\) in the form \(\frac { a } { b }\), where \(a\) and \(b\) are to be determined.
Aled decides to invest \(\pounds 1000\) in a savings scheme on the first day of each year. The scheme pays 8\% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by
$$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08 ^ { 2 } + 1000 \times 1 \cdot 08 ^ { 3 }$$
Calculate, to the nearest pound, the amount of savings at the end of thirty years.