Piecewise models and continuity Problem set B

Find a so that \(f\) is continuous at x = 3: \(f(x)=\begin{cases}2x+a & x<3\\ x^2-4 & x\ge3\end{cases}\).

Find a so that \(g\) is continuous at x = 2: \(g(x)=\begin{cases}ax^2+2 & x\le2\\ 3x-1 & x>2\end{cases}\).

Find a so that \(h\) is continuous at x = 4: \(h(x)=\begin{cases}x^a & 1\le x<4\\ 3x+6 & x\ge4\end{cases}\).

Find a so that \(f\) is continuous at x = 3: \(f(x)=\begin{cases}(x+1)^a & x<3\\ 3x^2+2 & x\ge3\end{cases}\).

Find a so that \(h\) is continuous at x = 3: \(h(x)=\begin{cases}e^{ax} & x<3\\ x^2+5 & x\ge3\end{cases}\).

Find a so that \(g\) is continuous at x = 3: \(g(x)=\begin{cases}\ln(ax) & x<3\\ 2x-4 & x\ge3\end{cases}\).

A tank holds water with 500 mg/l of a pollutant. Each day 9% of the pollutant is removed, so after n days the level is \(500\times0.91^n\) mg/l. Find the least whole number of days after which the level is less than 40 mg/l.

A phone plan costs, in pounds, \(C=12+0.9x\) for \(0\le x\le20\) GB, and \(C=ax+6\) for \(x>20\). The cost has no sudden jump at 20 GB. Find a.

The number of bacteria in a tank, in thousands, is \(900e^{-0.18d}\) after d days of treatment. Find the least whole number of days after which fewer than 50 thousand remain.

A car park charges £P for a stay of h hours as follows. Nothing for the first 2 hours. £2.50 for each hour after that, up to 5 hours in all. £4 for each hour after 5 hours.

1. (a) Find the charge for a stay of 7.5 hours.
2. (b) Find the length of stay that costs £27.50.

The function \(f\) is continuous at x = 3: \(f(x)=\begin{cases}e^{ax} & x<3\\ \ln(ax)+4 & x\ge3\end{cases}\). Find the two values of a.

A patient takes 300 mg of a drug. The amount in the blood has a half-life of 5 hours. Find the least whole number of hours after which less than 2 mg remains.