   Chapter 11.3, Problem 91E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Biology-Reproduction The Verhulst model for population growth specifies the reproductive rate of an organism as a function of the total population according to the following formula: R ( p ) = r 1 + k p , where p is the total population in thousands of organisms, r and k are constants that depend on the particular circumstances and the organism being studied, and R ( p ) is the reproduction rate in thousands of organisms per hour. k = 0.125  and  r = 45 , find R ' ( p ) andthen R ' ( 4 ) . Interpret the result.

To determine

To calculate: The derivative R(p) and then R(4) such that the reproductive rate of an organism is given by the function R(p)=r1+kp where p is the population in thousands of organisms, k=0.125 and r=45. Interpret the answers.

Explanation

Given Information:

The reproductive rate of an organism is given by the function R(p)=r1+kp where p is the population in thousands of organisms, k=0.125 and r=45.

Formula used:

Quotient rule of derivative of differentiable functions, f(x) and g(x) is,

ddx[f(x)g(x)]=f(x)g(x)f(x)g(x)[g(x)]2 where, g(x)0.

A derivative of a function using power rule y=xn is dydx=nxn1.

A derivative of a constant is 0.

Sum rule of the derivative is ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)] where f(x) and g(x) are any two differentiable functions.

Constant multiple rules of derivative of a function f(x) are ddx[cf(x)]=cddx[f(x)] where c is constant.

Calculation:

Consider the function,

R(p)=r1+kp

Substitute the value of k=0.125 and r=45 in the function.

R(p)=(45)1+(0.125)p=451+0.125p

Calculate the derivative of the function with respect to p.

Apply quotient rule,

R(p)=ddp(45)(1+0

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