   Chapter 8.2, Problem 88E

Chapter
Section
Textbook Problem

# Memory ModelA model for the ability M of a child memorize, measured on a scale from 0 to 10, is given by M = 1 + 1.6 t ln t ,   0 < t ≤ 4 where t is the child’s age in years. Find the average value of this model(a) Between the child’s first and second birthdays.(b) between the child’s third and fourth birthdays. (a)

To determine

To calculate:

Average value of the model M=1+1.6tlnt, 0<t4 on the scale of 1 to 10 between the child’s first and second birthdays.

Explanation

Given:

Model for the ability M of a child to memorize M=1+1.6tlnt,0<t4

Formula used:

Average value=1baabf(x)dx

Integration by parts:

udv=uv(vdu)

Calculation:

Consider the ability for memory of a child is modeled by a function:

M=1+1.6tlnt

Where, the child’s age t varies from 0 to 4.

Let’s calculate the average value of function y=M(t)=1+1.6tlnt when the child’s age t is in between 1 and 2.

The average value of y=f(x) in the interval (a,b) is given as,

Average value=1baabf(x)dx

Hence, the average value of this function y=M(t)=1+1.6tlnt in the interval (1,2) is

Average value=12(1+1.6tlnt)dt ……(1)

Applyintegration by parts formula, to integrate the integral tlntdt.

To apply integration by parts, write the integral in the form of udv.

Let u=lnt and dv=tdt, to get,

u=lntdu=1tdt

And

dv=tdtv=dv=tdt=t22

Now integrate the integral tlntdt by using integration by parts formula:

udv=uv(vdu)

(b)

To determine

To calculate: Average value of the model M=1+1.6tlnt,0<t4 on the scale of 1 to 10 between the child’s first and second birthdays

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