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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

In someone infected with measles, the virus level N (measured in number of infected cells per mL of blood plasma) reaches a peak density at about t = 12 days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of N(t) from t = 0 to t = 12 (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells × time). The function N has been modeled by the function

f ( t ) = t ( t 21 ) ( t + 1 )

Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.

Chapter 5.1, Problem 19E, In someone infected with measles, the virus level N (measured in number of infected cells per mL of

Source: J. M. Heffernan et al., “An In-Host Model of Acute Infection: Measles as a Case Study,” Theoretical Population Biology 73 (2006): 134–47.

To determine

The total amount of infection needed to develop symptoms of measles.

Explanation

Given information:

The virus level reaches the peak density at t=12days.

The function N is modeled by the function f(t)=t(t21)(t+21).

The subintervals is n=6.

Calculation:

Draw the graph of N(t) as shown in Figure 1.

Find the width (Δt) using the relation:

Δt=ban (1)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 12 for b, 0 for a, and 6 for n in Equation (2).

Δt=1206=2

The intervals are 0, 2 days, 4 days, 6 days, 8 days, 10 days, and 12 days.

The midpoints of the intervals are 1 day, 3 days, 5 days, 7 days, 9 days, and 11 days.

The expression to find the total amount of infection needed to develop symptoms of measles (Mn) is shown below:

Mn=f(t1)Δt+f(t2)Δt+...+f(tn)Δt (2)

Rearrange Equation (2) for the n=6 and mid points of the intervals as shown below:

M6=f(1)Δt+f(3)Δt+f(5)Δt+f(7)Δt+f(9)Δt+f(11)Δt (3)

Refer Figure 1

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