Question

Use a graphing calculator to find an exponential model that shows the relationship between the decades x since 1950 and the population y. Use 0 for 1950, 1 for 1960, .... How can you rewrite the exponential growth function in Exercise 3 to find the yearly growth rate?

a) By taking the derivative of the exponential function
b) By adding a constant to the exponential function
c) By multiplying the exponential function by the x-value
d) By subtracting a value from the exponential function

Answer 1

To find the yearly growth rate from an exponential model, manipulate the equation to express it in terms of e to find the continuous growth rate. The growth rate (r) is explicit in the formula P = Poer(t-to). Otherwise, log-transform the equation and solve for r if not already in this form.

Step-by-step explanation:

You asked how to rewrite the exponential growth function to find the yearly growth rate. To find the yearly growth rate from an exponential growth function, you do not take the derivative or add, multiply or subtract any constants directly within the function. Instead, you would typically manipulate the function to express it in terms of e (the base of natural logarithms) raised to a power that represents the continuous growth rate. For the equation of the form P = Poer(t-to), the r represents the yearly growth rate, and no further transformation is required. However, if your exponential function is not already in this form, you would log-transform the equation and then solve for the corresponding exponential rate.

For instance, if we have an equation y = ab^x and we want to express it in terms of e, we can take the natural logarithm of both sides and use properties of logarithms to isolate the growth rate. Remember that an exponential model is characterized by a constant percentage growth rate, which means as the value of the dependent variable (population, in this case) gets larger, its rate of growth also increases proportionally.