**Chapter 1 Project: Medicine in the Bloodstream**

**By: Tu Hoang**

Ms. Alexander

AP Calculus 2A

Ms. Alexander

AP Calculus 2A

Background Information

A patient's kidneys purify 25% of the blood in her body in 4 hours.

In this example, a patient takes one 16 milligram dose of a medication.

Our objective is to deteremine the amount of medication left in the patient's body after a certain amount of hours.

Question # 1

Determine the amount of medication left in the patient's body after 4, 8, 12, and 16 hours.

After 4 hours, there would be 12 milligrams left in her body.

After 8 hours, there would be 9 milligrams left in her body.

After 12 hours, there would be 6.75 milligrams left in her body.

After 16 hours, there would be 5.0625 milligrams left in her body.

Question # 2 & 3

# 2.

Use the information to write an equation that represents the of medication left in the patient's body after n 4-hour periods.

An equaton that represent the amount

"a"

, the medication left in the bloodstream in the patient's body after the intervals of 4-hours period,

"n"

, can be shown as

a=16*((¾))^n

Question # 3

Can you find a value of

"n"

for which

"a"

equals 0? Explain

There is no value for

(n)

at which

(a)

would equal zero. As

(n)

approaches infinity,

(a)

would approach 0, but won't equal zero

Question # 4-9

Exercises 4-9

The patients takes an additional 16-milliliter dose every 4 hours.

4. Determine the amount of medication in the patient’s body immediately after taking the second dose.

After taking the second dose, there would be 28-mg of medication in her body

Question # 5

5. What is happening to the amount of medication in the patient’s body over time?

After the third dose, there would be 37-mg medication left, and after the fourth, 43.75-mg would be left in the patient’s body. The amount of medication is increasing in her body overtime

Question # 6

# 6. Let x represent the number of hours and y represent the amount of medication in the patient’s body in milliliter.

X= Hours

Y= Medication in blood stream

Question # 7

This graph show the relationship between between the number of hours, x, and the amount of medication in milligrams, y, that would be in the patient's system.

# 7.

Use the graph in Exercise 6 to find the limits.

The limit as x approaches 4 from the left equals 12.

The limit as x approaches 4 from the right equals 28.

The limit as x approaches 12 from the left equals 27.75.

The limit as x approaches 12 from the right equals 43.75.

Question # 8

The function are discontinuous at 4, 8, 12, and 16, because left and right limits do not equal each other. For this reason, the amount of medication in her blood gets less and less within the four hours, dropping from 16 mg to 12 mg (left limit). However, because she will take another dosage of 16 mg at the 4 hour mark, there will actually be 12+16=28 mg of medication in her bloodstream (right limit).

Discuss the continuity of the function represented by the graph in Exercise 6. Interpret any discontinuities in the context of the problem

Question # 9

Write and solve an equation to find this amount

The equation that could be use is:

A= 16 - 16((3/4)^n)

n

being the amount of 4 hour intervals, and A being the amount to take

Because

'n'

would equal 1 for the first four hour period, we can plug in 1 for n and solve.

A=16-12, A=4.

The patient would need to take a dosage of 4 mg to keep a constant amount of medication.

Work