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

Question: Given an initial value of A = 120 and a decay constant of k = -0.05, construct the exponential decay function. Answer: f(t) = 120e^(-0.05t)

Step 1: Identify the Initial Value (A)

Determine the starting value of the exponential growth or decay function. This value, often denoted as A, is usually given in the problem statement. It represents the quantity at t = 0, or the beginning of our time frame.

Step 2: Determine the Growth or Decay Constant (k)

Identify the growth or decay constant of the function. This constant, symbolized by k, characterizes how rapidly the function increases (when k>0, growth) or decreases (when k<0, decay) over time. This constant is often given in the problem statement, and it can be provided as a percentage or a decimal value.

Step 3: Construct the Function

Using the values of A and k found in steps 1 and 2, construct the exponential growth or decay function in the form f(t) = Ae^{kt}. To do this, replace A with the initial value and k with the growth/decay constant. Keep in mind to use a plus sign in the exponent if k is positive (for growth) and a minus sign if k is negative (for decay). By following these steps, you will be able to formulate an exponential growth or decay function based on the two provided pieces of information: the initial value (A) and the growth/decay constant (k).