Question

In someone affected 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 x 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.

          In someone affected 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 x 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.

Show more…

Added by Alex T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 5

Instant Answer

Solved by Expert Linda Hand

07/16/2021

Step 1

The six subintervals are: $[0,2]$, $[2,4]$, $[4,6]$, $[6,8]$, $[8,10]$, $[10,12]$. The midpoints of these intervals are: $1$, $3$, $5$, $7$, $9$, $11$. Show more…

Show all steps

lock

Thanks for your feedback!

In someone affected 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 x 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.

Play audio

Feedback

Powered by NumerAI

Linda Hand and 73 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Recommended Videos

Recommended Textbooks

Transcript

00:01 Okay, we want to find the area under this curve from t equals 0 to t equals 12, and we're going to do it by making six intervals and then use the midpoints to find the height.

00:12 Okay, so first we want to find out how wide the intervals are, delta x.

00:19 So that will be 12 minus 0 divide by 6, which means they're two each.

00:25 Hopefully you're not surprised by that.

00:28 Okay, so then here's what it would look like.

00:31 0, 2, 4, 6, 8, 10, 12.

00:39 Okay, i did pretty good that time.

00:41 Usually i run out.

00:43 Okay, and the reason i want to do that is because what we're interested in is the midpoints of those.

00:49 So the midpoints are 1, 3, 5, 7, 9, and 12.

00:56 No, 11, sorry.

01:01 Okay, so now what we're going to do is we're going to plug all those midpoints.

01:04 Points into this function and find out what its value is.

01:08 So what we're finding is a rectangle, really, whose width is two, and its height is f of whatever you have here.

01:20 Then we're going to just add up all those rectangles.

01:24 Okay.

01:25 So f of 1 will be 1 times 20 times is that t plus 1.

01:33 I better write that down again here...

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later