Chapter 4.3, Problem 68E

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640

Chapter
Section

Applied Calculus for the Manageria...

10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem

CONCENTRATION OF A DRUG IN THE BLOODSTREAM The concentration (in milligrams per cubic centimeter) of a certain drug in a patient’s bloodstream t In after injection is given by C ( t ) = 0.2 t t 2 + 1 Sketch the graph of the function C, and interpret your results.

To determine

To sketch: The graph of the function C and interpret the result.

Explanation

Given:

The concentration of certain drug in a patient’s bloodstream t hr after injection is given as,

C(t)=0.2tt2+1 (1)

Calculation:

As the function is given in rational form, thus the vertical asymptote for a rational number is the point where the function is undefined, that is the denominator is zero.

Since the denominator of the given function is never zero, thus there is no vertical asymptote for the given function.

The formula to calculate the horizontal asymptote (ie, the line y=b ) is either, limxf(x)=b or limxf(x)=b .

Take limits t tends to and use both sides of equation (1).

limtC(t)=limt(0.2tt2+1)=limt(0.2tt2t2(1+1t2)t2)[DivideNumeratorandDenominatorbyt2]=limt(0.2t(1+1t2))=0.2(1+1)

Further solve the above equation.

limtC(t)=01+0=0

Take limit t tends to and use both sides of equation (1).

limtC(t)=limt(0.2tt2+1)=limt(0.2tt2t2(1+1t2)t2)[DivideNumeratorandDenominatorbyt2]=limt(0.2t(1+1t2))=0.2(1+1)

Further solve the above equation.

limtC(t)=01+0=0

Since limtC(t)=0 and limtC(t)=0 , thus the horizontal asymptote is y=0 .

Differentiate equation (1) with respect to t.

C(t)=(t2+1)ddt(0.2t)(0.2t)ddt(t2+1)(t2+1)2=(t2+1)(0.2)(0.2t)(2t)(t2+1)2=0.2t2+0.20.4t2(t2+1)2=0.20.2t2(t2+1)2 (2)

This shows that for C(t)=0 , the critical point comes out to be,

0.20.2t2=00.2t2=0.2t2=1t=1

Thus, the intervals are (0,1) and (1,) .

Take the test points for different intervals in the table below to determine the signs of C(t) .

 Intervals Test point c C′(c) Sign of C′(t) (0,1) 0.5 0.096 + (1,∞) 3 −0.016 −

Thus, the function decreases on the interval (1,) and increases on the interval. (0,1)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Solve the equations in Exercises 126. 14x2=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In problems 63-73, factor each expression completely. 67.

Mathematical Applications for the Management, Life, and Social Sciences

Find the limit or show that it does not exist. limx1+4x62x3

Single Variable Calculus: Early Transcendentals

Given: mRST=39 mTSV=23 Find: mRSV Exercises 1624

Elementary Geometry for College Students