Question

Derivatives of Exponential Functions (Scenario)

Consider the following problem: Suppose that F(x) computes the rabbit population on a game reserve that doubles every 6 months. Suppose there were 200 rabbits initially.

1. Write a mathematical expression for F(x).

2. Find domain and range of F(x) and discuss its meaning

3. Find F(x) and F'(x) at any point and discuss its physical meaning

4. Find all x values for which F'(x)=0 and discuss what this function means

5. Discuss if your function F(x) is differentiable and why. If it is not, select another function that is and discuss the change you made.

6. Discuss the criteria for selecting a real-world scenario that would change if you were seeking to model it with a logarithmic function instead. What key similarities and differences would you find?

***Please discuss why...I am having a hard time understanding each meaning with these scenarios*** Thank You!!!

Answer 1

Solution-

The function can be written in double the 6 month interval system,

N=6n,

Here, N is total number of month , n is multipier for 6 month and f(0)=200,

(a) The mathametical expression for the scenarios is,

(b) Domain for the function is defined as number of values the function can satisfy with,

here, we only know the value of this funciton at N=6n so the function can only take N = [0,6,12,18,................6n)

n is natural number N=6n

Range of the function is sat of all output values,

here, we only get R=200X2N/6 or,

Range = [200,400,800,............,200X2N/6)

(c) Take N=18 months

Find the diffrentiation,

This shows after 18 months the growth rate for the rabbites will be 184.84 increment per month

(d) The rate of growth will never become zero as the f(N) is the exponential type function,

The slope will increase with increase in N,

No value of N exists.