Piecewise models and continuity Problem set A

Find a so that \(f\) is continuous at x = 2: \(f(x)=\begin{cases}3x+a & x<2\\ x^2-5 & x\ge2\end{cases}\).

Find a so that \(g\) is continuous at x = 3: \(g(x)=\begin{cases}ax^2-1 & x\le3\\ 2x+5 & x>3\end{cases}\).

Find a so that \(h\) is continuous at x = 3: \(h(x)=\begin{cases}x^a & 1\le x<3\\ 5x+5 & x\ge3\end{cases}\).

Find a so that \(f\) is continuous at x = 4: \(f(x)=\begin{cases}(x+1)^a & x<4\\ 2x^2+3 & x\ge4\end{cases}\).

Find a so that \(h\) is continuous at x = 2: \(h(x)=\begin{cases}e^{ax} & x<2\\ x^2+4 & x\ge2\end{cases}\).

Find a so that \(g\) is continuous at x = 4: \(g(x)=\begin{cases}\ln(ax) & x<4\\ 3x-10 & x\ge4\end{cases}\).

A sample of a radioactive substance has mass 80 g. Each day 6% of the mass decays, so after n days the mass is \(80\times0.94^n\) g. Find the least whole number of days after which the mass is less than 10 g.

A taxi fare in pounds for a journey of d km is \(F=3+1.6d\) for \(0\le d\le8\), and \(F=ad+7\) for \(d>8\). The fare has no sudden jump at 8 km. Find a.

The mass of a drug in a pond, in mg, is \(120e^{-0.07d}\) after d days. Find the least whole number of days after which less than 15 mg remains.

A country charges income tax £T on an income of £x as follows. No tax on the first £12 000. 20% of the income between £12 000 and £50 000. 40% of any income above £50 000.

1. (a) Find the tax on an income of £65 000.
2. (b) Find the income that pays £15 000 in tax.

The function \(f\) is continuous at x = 2: \(f(x)=\begin{cases}e^{ax} & x<2\\ \ln(ax)+3 & x\ge2\end{cases}\). Find the two values of a.

A radioactive sample has mass 640 g and a half-life of 11 days. Find the least whole number of days after which less than 4 g remains.