GroupHomework #1

A U.S. taxpayer pays no federal tax on the first $15,000 of annual income, 15% on the next $25,000 of income and 28% on the all earnings above $40,000. Let x denote a taxpayer's income (in dollars) and let T(x) denote the tax owed (in dollars) at that income level.

You may want to learn about piecewise continuous functions.

a) Compute T(10000), T(20000), T(30000), T(40000), and T(50000).b) Give a piecewise defined formula for T(x) and draw the graph of T(x) for NiMzMSIiIUkieEc2IjFGJiImKysn. Verify that when you use this formula for the values 10000, 20000, 30000, 40000 and 50000 your answers match your answers from part (a). c) If a taxpayer earning $56,789 earned one more dollar of income, the tax owed would increase by 28 cents. If a taxpayer earning $34,567 earned one more dollar of income, the tax owed would increase by 15 cents. The first taxpayer has a marginal tax rate of 28%, while the second taxpayer has a marginal tax rate of 15%. Let MR(x) denote the marginal tax rate (as a decimal) at an income level of x. Give a piecewise defined formula for MR(x). d) Draw the graph of MR(x) for NiMzMSIiIUkieEc2IjFGJiImKysn. Describe the relation between T(x) and MR(x) in your own words using complete sentences.

A spherical tank has a diameter of 10 feet. Water is flowing into the tank at a constant rate of 4 cubic feet per minute. a) How long does it take to fill the tank of water? Let D(t) be the depth of water in the tank t minutes after water begins flowing into the tank. b) What is the Range of values for D(t)?c) Is the graph of D(t) increasing or decreasing? Explain?d) Where is the graph of D(t) concave up and where is the graph concave down? Explain why this happens?

The Heaviside function is defined as NiMvLUkiSEc2IjYjSSJ0R0YmLUkqUElFQ0VXSVNFR0YmNiQ3JCIiITJGKEYtNyQiIiIyRi1GKA== is 0 for negative values and 1 for positive values. It frequently is used in engineering to represent a sudden change.a) Plot the Heaviside function for a suitable range of t.b) Find a function NiMtJkkiRkc2IjYjIiIiNiNJInRHRiY= of the form NiMvLSZJIkZHNiI2IyIiIjYjSSJ0R0YnKiZJIkFHRidGKS1JIkhHRic2IywmRitGKUkiY0dGJ0YpRik=, which is 0 for t < 3 and 5 for t > 3. You will need to find the appropriate values for A and c.c) Find a function NiMtJkkiRkc2IjYjIiIjNiNJInRHRiY= of the form NiMvLSZJIkZHNiI2IyIiIzYjSSJ0R0YnLCZJIkJHRiciIiIqJkkiQUdGJ0YuLUkiSEdGJzYjLCZGK0YuSSJjR0YnRi5GLkYu, which is 2 for t < 1 and 1 for t > 1. You will need to find the appropriate values for A, B and c.d) Find a function NiMtJkkiRkc2IjYjIiIkNiNJInRHRiY= of the form NiMvLSZJIkZHNiI2IyIiJDYjSSJ0R0YnLCYtSSJmR0YnRioiIiIqJi1JImdHRidGKkYvLUkiSEdGJzYjLCZGK0YvSSJjR0YnRi9GL0Yv, which is NiMqJCwmSSJ0RzYiIiIiIiIjISIiRig= for t < 2 and NiMsJiIiIyIiIkkidEc2IiEiIg== for t > 2. You will need to find the appropriate value for c and appropriate functions f(t) and g(t).

A function which grows exponentially is generally written in the form NiMvLUkiZkc2IjYjSSJ0R0YmKiYmSSJ5R0YmNiMiIiEiIiIpSSJlR0YmKiZJImtHRiZGLkYoRi5GLg==. This question is to show that an exponential function can always be written in this form.a) Find values of NiMmJSJ5RzYjIiIh and NiMlImtH so that NiMvKSIiIyUidEcqJiYlInlHNiMiIiEiIiIpJSJlRyomJSJrR0YsRiZGLEYs.

NiMmSSJ5RzYiNiMiIiE= is the value of NiMtSSJmRzYiNiMiIiE=.

b) Find values of NiMmJSJ5RzYjIiIh and NiMlImtH so that NiMvKSIiJComIiIlIiIiJSJ0R0YoKiYmJSJ5RzYjIiIhRigpJSJlRyomJSJrR0YoRilGKEYo.c) Find values of NiMmJSJ5RzYjIiIh and NiMlImtH so that NiMvKSIiJiwmJSJ0RyIiIiIiI0YoKiYmJSJ5RzYjIiIhRigpJSJlRyomJSJrR0YoRidGKEYo.d) Find values of NiMmJSJ5RzYjIiIh and NiMlImtH so that NiMvKiYlIkFHIiIiKSUiQkcsJiomJSJDR0YmJSJ0R0YmRiYlIkRHRiZGJiomJiUieUc2IyIiIUYmKSUiZUcqJiUia0dGJkYsRiZGJg==. These values will depend on A, B, C and D