PreCalculus
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1. In the United States, payroll taxes are levied to fund the Social Security Insurance and Medicare Insurance programs. The yearly tax is approximately as follows (round up )
(i) On all income, pay 1.5% toward Medicare Insurance.
(ii) Pay 6% on any income up to $100,000, toward Social Security Insurance.
(iii) Pay no additional tax toward Social Security, on any income beyond $100,000.
(I), make a copy of the following table, and then fill it out. Show your work!
20,000 75,000 100,000
500,000
(**) This means you will have to do a footnote in #4 when you look up the current Senatorial salary. (In addition, this assumes that a person with such salary pays these taxes. Actually, public employees, such as US Senators and UMass professors, do not pay into Social Security, but rather into other government plans. But ignore that for here, and just get the Math right.)
(II) If x is the income earned, write the piecewise formula for the total tax paid for these social insurance programs.
(III) Make a graph of (II). Be sure to label it clearly. In particular, this means to write the coordinates at the “joints” of this function. Again, be sure to use a straight edge and graph paper, and label clearly the UNITS of each axis.
Yearly Income $$
Tax paid toward Medicare
Tax Paid toward Social Security
Total Tax Paid
Percentage of Income paid to Insurance
A US Senator (**) see note below
(IV) Use (III) to find the income one would pay if his total tax were each of the following. Be sure to show all of your algebra steps.
(a) $6000; (b) $10,000;
(V) Next, use your answer from (II) to find a formula for P (x), the percentage tax a worker pays. Be sure to write this as a piecewise formula.
(VI) Show that your formula in (V) matches the answers you found in the table above.
(VII) Now find the limit as x!+ ∞for P (x).
(VIII) Now draw a clearly labeled sketch of the function you found in (V). Be sure to
indicate clearly the horizontal asymptote, based upon the work you did in (VI).
(IX) Explain why a person who knows no mathematics at all, but who has good common sense, could have answered question (VII). Do so by explaining the real meaning of this question. You might use terms like “Jeff Bezos” or “Bill Gates” to
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make your point.
2. (6 points) Now go back to the P (x) you found in 1(V).
(I) Find P ‘ (x), the derivative of that piecewise function, BY USING THE 4-STEP DEFINiTION slope quotient. [No credit for using derivative rules.] Note that your derivative function will likewise be piecewise, so you will have to apply the derivative definition to each “piece”.
(II) Now check your answer in (I) by applying the derivative rules. Does your answer agree with (I)?
(III) For the US Senator in #1, find the value of
P’ (x).
(III) Now in complete sentences, explain what (III) means. Your answer must tell us the RATE, in ???? S per ? Use words like “decreases” or “increases”.
3. (5 pt) Consider the ratio function f(x) = (x2-9)/ (x2 + 2x).
(I) Explain the DOMAIN of this function. Show all work necessary to find the “forbidden” values. [No credit just for stating the answer; you must provide work] and then write your final answer in INTERVAL NOTATION. If you are not sure what that means, be sure to look it up first, and cite your reference in the bibliography of problem #4.
(II) Use (i) to find the equations of the vertical asymptotes. (III) Now find the limits as x goes to +/- infinity.
(IV) Use (iii) to write the equation(s) of any horizontal asymptote(s). Be sure to present the final answer in a complete sentence.
(V) Now draw axes (with a straightedge on graph paper), and sketch the asymptotes you found in (II) and (IV). Be sure to label them! The term “LABEL” means to draw the lines and then write their equations on your sketch with each equation appearing in a location that make it obvious what is being labeled.
(VI) Now find the limits as x goes to (i) -2- ; (ii) -2+; (iii) 0-; (iv) 0+. As we did in class, use an appropriate nearby value to show how you find each limit. Again, no credit without showing work.
(VII) Find all ZEROES of this function. List them as ordered pairs. Show all your work, and then present your final answer in a complete sentence.
Project ID: #29580047
