Problem 2 - Calculus Early Transcendentals

Question

Give two pieces of information that may be used to formulate an exponential growth or decay function.

Step-by-Step Solution

Answer

Answer: The two essential pieces of information required for formulating an exponential growth or decay function are 1) the initial value (A) and 2) the growth/decay rate (r). The initial value represents the starting point of the function and affects the overall scale of the graph, while the growth/decay rate determines the steepness and direction of the curve on a graph and affects how quickly the quantity or population size changes over time.

Step 1: 1. Initial value#a

The initial value is the starting point of the exponential function. In an exponential growth or decay function, it represents the quantity or population size at time t=0. It is commonly denoted as A (for "amount"). For example, if we have a population of bacteria that has 1000 individual bacteria initially, A = 1000. This value is the base of the function and affects the overall scale of the graph.

Step 2: 2. Growth/decay rate#a

The growth or decay rate is another critical piece of information needed to formulate an exponential growth or decay function. It represents the rate at which the quantity or population size is increasing (growth) or decreasing (decay) over time. It is commonly denoted as r (for "rate"). In exponential growth functions, the rate is a positive value, while in exponential decay functions, the rate is a negative value. The rate determines the steepness and direction of the curve on a graph, and it affects how quickly the quantity or population size changes over time. For instance, a higher growth rate results in a steeper curve and a more rapid increase in the population size. Given these two pieces of information, we can formulate an exponential function as follows: $$f(t) = A \cdot e^{r \cdot t}$$, where f(t) is the quantity or population size at time t, A is the initial value, r is the growth/decay rate, and t is the time.