Week5a

Clicker Questions

• Piecewise functions
• Intermediate Value Theorem

Activities

Activity: Income tax

In 2013, Canadian government established the following federal tax rates:

• 15% on the first $\$43,561$ of taxable income
• 22% on the next $\$43,561$ of taxable income
• 26% on the first $\$43,931$ of taxable income
• 29% on taxable income over $\$135,054$

Questions:

1. Sketch a graph representing the tax rate as a function of taxable income.
2. Define a piecewise function representing the amount of tax owned as a function of taxable income. Is this function continuous?
3. Sketch the graph of the function defined in 2.
4. Canada's income tax rates are called progressive because the rate increases as income increases. Suppose the government decided to apply a flat tax rate of 25% to everyone. For which income(s) would this flat rate of 25% be the same as the current progressive tax? Give a geometrical explanation.

Extra discussion questions

1. Find a function that has at least one root between 0 and 1.
2. Explain why any odd-degree polynomial has at least one root.
3. Show that $\sqrt{2}$ is between 1 and 2.