Tax rates and concavity In this activity, we will use some simple ideas about taxes and tax rates to investigate some basic properties about functions. Economists use the term marginal tar rate to refer to the additional tax paid on the next (taxable) dollar earned. For example, suppose that under some tax plan the tax on $34,567 is $6897.76 while the tax on $34,568 is $6898.04. Then, for a taxpayer earning $34,567, the tax paid on the next dollar is $0.28, so the marginal tax rate is $0.28/$1 = 28%. (Note that this is not directly related to $6897.76/$34,567 = 19.95%, the percentage of taxable income that goes to tax.) Let t(x) denote the amount of tax that is paid on $x of taxable income. 1. Economists refer to a tax as progressive if the marginal tax rate increases (as taxable income increases). If t(x) is a progressive tax, what will the graph of t(x) look like? Sketch t(x) for a progressive tax. 2. At a few points on your graph, sketch the tangent line. What is the relationship between the graph and its tangent lines? 3. Explain how the marginal tax rate is related to t'(x). (Hint: Take Ac = $1 in the definition of t'(x).) 4. If t has a second derivative, what property will t" have? Tax rates and concavity In this activity, we will use some simple ideas about taxes and tax rates to investigate some basic properties about functions. Economists use the term marginal tar rate to refer to the additional tax paid on the next (taxable) dollar earned. For example, suppose that under some tax plan the tax on $34,567 is $6897.76 while the tax on $34,568 is $6898.04. Then, for a taxpayer earning $34,567, the tax paid on the next dollar is $0.28, so the marginal tax rate is $0.28/$1 = 28%. (Note that this is not directly related to $6897.76/$34,567 = 19.95%, the percentage of taxable income that goes to tax.) Let t(x) denote the amount of tax that is paid on $x of taxable income. 1. Economists refer to a tax as progressive if the marginal tax rate increases (as taxable income increases). If t(x) is a progressive tax, what will the graph of t(x) look like? Sketch t(x) for a progressive tax. 2. At a few points on your graph, sketch the tangent line. What is the relationship between the graph and its tangent lines? 3. Explain how the marginal tax rate is related to t'(x). (Hint: Take Ac = $1 in the definition of t'(x).) 4. If t has a second derivative, what property will t" have