- In order to write a function representing the amount of dose that remains, we must define what is already known and the type of function that we will be modifying to find the amount of insulin remaining in the body.
- We know that first the body weight adjusted dose is generally 12 units and that the insulin breakdown in the body is 4.9% each minute. The fact that a percentage is used to represent the rate of breakdown signifies that this will not be a linear representation of insulin breakdown but that exponential decay will be used to represent such insulin breakdown.
- The skeleton of our equation that we will be using to model the insulin breakdown will be the equation A=Pe^rt known as the "Pert" equation. In the "Pert" equation, A represents our output, P represents our starting value which will become bigger or smaller. How it becomes bigger or smaller depends on our value for r which is the rate at which exponential growth or decay occurs. Lastly, t represents the time in whatever units you are given.
- By applying the definitions of the Pert equation, We see we are trying to solve for A since P is given to us as 12 units since that is the adjusted dose which is our starting value. r will be -4.9% which is equal to -0.049 and r is negative since the adjusted dose is decreasing be 4.9% a minute. Our variable time t is also known but this is fine since it acts as our independent variable or our input to generate our output A.
- Now to put it all together, A=Pe^rt=12e^0.049t. This here is our function that will represent our breakdown of insulin by 4.9% each minute starting with the adjusted dose of 12 units.
