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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Memory Model A model for the ability M of a child to memorize, measured on a scale from 0 to 10, is

M = 1 + 1.6 t ln t , 0 < t 4 where t is the child’s age in years.

(a) Find the average value of this model between the child’s first and second birthdays.

(b) Find the average value of this model between the child’s third and fourth birthdays.

(a)

To determine

To calculate: The average value of the model between child’s first and second birthday when the model for the ability M of the child to memorize and measured on the scale from 0 to 10 which is given by the function,

M=1+1.6t lnt,  0<t4

Where t is the age of child in years.

Explanation

Given Information:

The function,

M=1+1.6t lnt,  0<t4

Formula used:

Integration by parts.

When u and v is assumed to be the differentiable functions of x then,

u dv=uvv du

Calculation:

Consider the given data and the function,

M=1+1.6t lnt,  0<t4

In order to calculate the average value of the model between child’s first and second birthday integrate the function M=1+1.6t lnt with lower limit as 1 and upper limit as 2.

Thus,

Average value=12121(1.6t ln t+1) dt=21dt+1.621(t ln t) dt

In order to obtain the value of the integral 1.621(t ln t) dt apply integration by parts formula,

Let u=lnt and dv=t dt, then

dv=t dt

Apply integral on both sides of the above equation as,

dv=t dtv=t22

Thus, v=t22,

Then differentiate both sides of the equation u=lnt as,

du=d(lnt)du=1t dt

So, du=1t dt

Now apply integration by parts formula.

Substitute u=lnt, dv=t dt, v=t22 and du=1t dt in the formula u dv=uvv du as,

(lnt)t dt=(lnt)(t2

(b)

To determine

To calculate: The average value of the model between child’s third and fourth birthday when the model for the ability M of the child to memorize and measured on the scale from 0 to 10, is given by the function,

M=1+1.6t lnt,  0<t4

Where t is the age of child in years.

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