   Chapter 9.5, Problem 56E Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

Alzheimer's As the baby boomer generation ages and the proportion of the U.S population over 65 increases, the number of Americans with Alzheimer's disease and other dementia is projected to grow each year. With data from the Social Security Administration and the Alzheimer’s Association for selected years from 2000 and projected to 2050, the percent of the U.S. population with Alzheimer’s can be modeled with the function A ( t ) = 0.34 t 2 + 1.3 t + 450 − 0.012 t 2 + 3.3 t + 260 where t is the number of years past 1990.(a) Find the function that gives the instantaneous rate of change of A(t).(b) Use your answer in part (a) to find and interpret the instantaneous rate of change of the percent of the U.S. population with Alzheimer’s in 2025.

(a)

To determine

To calculate: The function for the instantaneous rate of change of Alzheimer’s disease A(t), if the baby boomer generation ages and the proportion of the U.S. population over 65 increases, the number of Americans with Alzheimer’s disease and other dementia is projected to grow each year.

Explanation

Given Information:

The model function for the percent of the U.S. population with Alzheimer’s disease by Social Security Administration and the Alzheimer’s Association for selected years from 2000 and projected to 2050 is A(t)=0.34t2+1.3t+4500.012t2+3.3t+260.

Formula Used:

The quotient rule for the derivative of the two function, ddx(fg)=gdfdxfdgdxg2.

The sum and difference rule of derivate of functions, ddx[u(x)±v(x)]=ddxu(x)±ddxv(x).

The simple power rule of derivative ddx(xn)=nxn1.

Calculation:

Consider the function for the percent of the U.S. population with Alzheimer’s disease, A(t)=0.34t2+1.3t+4500.012t2+3.3t+260.

Differentiate the provided model function,

dA(t)dt=ddt[0.34t2+1.3t+4500.012t2+3.3t+260]

Use the quotient rule for the derivative of the two function f(x) and g(x) is, ddx(fg)=gdfdxfdgdxg2.

dA(t)dt=(0.012t2+3.3t+260)ddt(0.34t2+1.3t+450)(0.34t2+1.3t+450)ddt(0.012t2+3.3t+260)(0.012t2+3.3t+260)2

Use the sum or difference rule of derivative, ddx[u(x)±v(x)]=ddxu(x)±ddxv(x)

(b)

To determine

To calculate: The instantaneous rate of change of the percent of the U.S population with Alzheimer’s disease in 2025 and estimate its interpretation also, if the baby boomer generation ages and the proportion of the U.S. population over 65 increases, the number of Americans with Alzheimer’s disease and other dementia is projected to grow each year. With data from the Social Security Alzheimer’s Association for selected years from 2000 and projected to 2050, the Alzheimer’s can be modeled by A(t)=0.34t2+1.3t+4500.012t2+3.3t+260 where t is the number of years past 1990.

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