Exponential Growth & Decay Equations & Examples

Recent questions in Exponential growth and decay
Algebra IAnswered question
sombereki51 sombereki51 2022-09-22

A colleague came across this terminology question.
What are the definitions of exponential growth and exponential decay? In particular:
1) Is f ( x ) = e x exponential growth, decay, or neither?
2) Is g ( x ) = e x exponential growth, decay, or neither?
Consider f ( x ) = A e k x .. I can't find any sources that specify A>0. My answer is that:
f exhibits
1. exponential growth for A>0,k>0, and
2. exponential decay for A>0,k<0
whereas | f | exhibits
3. exponential growth for A<0,k>0, and
4. exponential decay for A<0,k<0.
In case (3) we shouldn't call f an exponential growth function without noting that it is "negative growth". Also it wouldn't be called it an exponential decay function without specifying the "direction of decay", so it is neither.
In case (4) it's neither as well. One should specify that it is the magnitude of f which decays exponentially although f is increasing in value. Although f is increasing in value, is it growing? It seems odd to say it is exponentially growing.
It just doesn't sit right with me to refer to a function as growing if it is decreasing in value. Certainly, it's magnitude may be growing.
Next consider a function with exponential asymptotic behavior (e.g. logistic) so that as x , f ( x ) A e k x + C for some k>0. I feel the best way to describe this would be "exponential decay towards C" with a qualification as being from as being from above or below depending on the sign of A.
If someone is to just use the terminology "exponential growth (decay)", it implies f ( x ) = A e k x with positive A and k>0 (k<0) unless there is a specific context or further clarification as to what the actual nature of the function is.

Algebra IAnswered question
tuzkutimonq4 tuzkutimonq4 2022-09-11

Consider the single species population model defined by
d R d t = g R R + R m d R ,
for t>0, where g , R m , and d are all positive parameters and R ( 0 ) = R 0
(a) Describe the biological meaning of each term in the equation.
(b) Determine the steady-states of the system and discuss any constraints on the model parameters for the model to admit biologically meaningful solutions.
(c) Determine the steady-state stability and discuss any variation in this with respect to the model parameter values.
a) gR represents the exponential growth of population, dR represents the exponential decay of population, g is the growth rate, and d is the decay rate. What do R and R m represent? How can I define the term
g R R + R m ?
What is R + R m ? Does it affect the gR for the grow?
b) In single steady-state system,
d R d t = 0.
g R R + R m d R = 0
g R d R ( R + R m ) = 0
R ( g d ( R + R m ) ) = 0
either R=0 or g d ( R + R m ) = 0
g d R m = 0       ( R = 0 )
R m = g d
R = g d       ( R m = R )
So, we have R 1 a s t = 0, R 2 = g d Are these correct? I am not sure constraint on the model parameters to admit biologically meaning solutions?
c) to determine steady-state stability let
f ( R ) = g R R + R m d R
d f d R = g l n ( R + R m ) d .
My differentiation may be wrong and don't know the term R m while differentiating with respect to R. I really don't know after that. and I know my answer is still incomplete.

Exponential growth and decay subject related to one of the more complex aspects of Algebra, which makes it relatively difficult for students to cope with it as it requires analysis and knowledge of the basics. Take your time to explore various exponential growth and decay practice answers below to refresh your memory and see some helpful examples.

The answers that you can see below must be linked to the questions to see the reasons why certain solutions have been provided. Remember that analysis will be helpful to see the correct approach!