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LC4 Math Project

Introduction
Calculus is used in every branch of the physical sciences, actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other.

TASK 1
The spread of a virus is modeled by V(t) = -t2 + 6t -4, where V(t) is the number of people (in hundreds) with the virus and t is the number of weeks since the first case was observed.

QI: When does the number of cases reach maximum?

When the derivative of the function is equals to 0.

QII: What is the maximum number of cases?

The derivative of V(t) = -t2 + 6t -4 is equals to V'(t) = -2t+6.

As V'(t)=0 when the number of cases reach to maximum, so :-

V'(t) = 0 = -2t+6 	-2t = 0 – 6 	                 t = (-6)/(-2)   	t = 3

So :- the maximum number of cases is :

V(t) = -t2 + 6t -4                  V(3) = -32 + 6(3) -4 = 5                    5×100 = 500 cases is the maximum.

After three weeks of the spreading of the disease, the number of victimized people by the disease reaches to the maximum number of influenced people which is 500 cases.

QIII: The concentration of a drug in a patient’s bloodstream h hours after it was injected is    given by

Find the concentration of the drug after:

a. 5 hours                       b. 10 hours                             c. 15 hours

A(h)(0.17(5))/(5^2+2) = 0.031             b)  A(h)(0.17(10))/(〖10〗^2+2) = 0.016               c)  A(h)(0.17(15))/(〖15〗^2+2) = 0.011

Interpret the concentration of the drug in the bloodstream after a really long time. Explain. after a long time, the concentration of the drug in the bloodstream is going to approximately vanish. Because as the number of hours increase the concentration decrease.

Find and confirm your interpretation. 〖limA(h)〗┬(h→∞)⁡〖(0.17h/(h^2+2))^1 〗 as shown in this formula, as far as the limit is going to infinity :

(0.17(∞))/(∞^2+ 2) = 0, which means that as the limit goes to infinitive number of hours, the drug concentration keeps decreasing till it reaches to an approximate vanish.

TASK 2 A simplified income tax considered in the U.S. senate in 1986 had two tax brackets. Married couples earning $29300 or less would pay 15% of their income in taxes. Those earning more than $29300 would pay $4350 plus 27% of the income over $29300 in taxes. Let T(x) be the amount of taxes paid by someone earning x dollars in a year. QI:

lim┬(x⟶〖29300〗^- )⁡〖T(x)〗 lim┬(x→〖29300〗^- )⁡〖(0.15x)^1 〗 = 0.15(29300) =4395

lim┬(x⟶〖29300〗^+ )⁡〖T(x)〗 lim┬(x→〖29300〗^+ )⁡〖(4350+0.27(x-29300)^1 〗 = 4350+0.27(29300-29300) =4350

lim┬(x⟶29300)⁡〖T(x)〗 lim┬(x→29300)⁡〖T(x)〗 = Does Not Exist, because the limit from right does not equal the limit from left. lim┬(x→〖29300〗^+ )⁡〖T(x)〗 ≠ lim┬(x→〖29300〗^- )⁡〖T(x)〗 	4350 ≠ 4395

QII. Sketch the graph of T(x)

b) Identify any x-values where T is discontinuous T is discontinues at x = 29300.

QIII. Let A(x)=(T(x))/x  be the average tax rate, that is the Amount paid in taxes divided by the income.

Find a formula for A(x). (Note the formula will have two parts: one for x≤29300 and one for x>29300) 0.15x/x                                          x ≤ 29300 (4350+0.27(x-29300))/x                     x > 29300 Find lim┬(x⟶〖29300〗^- )⁡〖A(x)〗 〖limA(x)〗┬(x→〖29300〗^- )⁡〖(0.15x/x)^1 〗 = (0.15(29300))/29300 = 0.15

Find〖 lim〗┬(x⟶〖29300〗^+ )⁡〖A(x)〗 〖limA(x)〗┬(x→〖29300〗^+ )⁡〖((4350+0.27(x-29300))/x)^1 〗 = (4350+0.27(29300-29300))/29300 = 0.148

Find〖 lim〗┬(x⟶29300)⁡〖A(x)〗 〖limA(x)〗┬(x→〖29300〗^+ )⁡〖((4350+0.27(x-29300))/x)^1 〗 ≠ 〖limA(x)〗┬(x→〖29300〗^- )⁡〖(0.15x/x)^1 〗     so:- 〖 lim〗┬(x⟶29300)⁡〖A(x)〗 = does not exist.

Find and Interpret lim┬(x⟶∞)⁡〖A(x)〗

lim┬(x⟶∞)⁡〖A(x)〗 = 〖limA(x)〗┬(x→〖29300〗^+ )⁡〖((4350+0.27(x-29300))/x)^1 〗 ( because lim┬(x⟶∞)⁡〖A(x)〗 is from the right side ( ∞+).

lim┬(x⟶∞)⁡〖A(x)〗 (4350+0.27(x-29300))/x  = 0.27x/x = (0.27(∞))/∞ = 0.27

We concluded, whatever the income is, the average tax rate is constant.