Question

How do you find the exponential growth function that models a given data set?

Answer 1

Hint: To solve this problem, first of all we need to have a clear cut idea of the underlying concept and understand what the general equation for an exponential function looks like. The general equation for any exponential function that models a given data set is given by,
\[y=a{{e}^{bx}}+c\] Where ‘a’, ‘b’ and ‘c’ are unknown parameters that need to be calculated. The main motive or objective behind this, is to find out all the unknown variables in the function and then put them back to the function and to find data for any given value of ‘x’.

Complete step by step solution:
Now we start off with the solution to the given problem by writing that this problem requires some given initial conditions which are very necessary to find the values of the unknown parameters of the equation \[y=a{{e}^{bx}}+c\] . The unknown variables in this equation are ‘a’, ‘b’ and ‘c’. There must be at least three initial conditions given so as to find the value of the three unknown parameters. After finding the value of the parameters we need to put these values to the original equation. Now we can put data set values in the formed equation to find out the required parameter ‘y’. This equation now becomes a generalised equation to find all the possible values for a given set of data.

Note: Solving these types of problems can sometimes be very tedious when the number of unknown parameters given shall be huge. Finding all the values of the parameters will lead us to the final formed equation which can be used for a wide range of dataset. To solve such problems more efficiently, we need to have a good grasp of topics like linear equations, data modelling and linear progressions.