Module 15

What you need to learn (Learning Objectives):
In this module we learn that psychological processes (such as human memory) can be modeled by mathematical equations.
You will learn to solve these equations.

Why do you need to learn this?
In real life, we often need to find mathematical equations that can model psychological processes and we need to be able to solve such equations in order to track the course of such processes.

Vocabulary:
Logarithmic function, natural logarithmic function, modeling.

Model:
Logarithmic functions are useful in modeling data that increases or decreases slowly. What do we mean by a logarithmic function? The function f(x) = log_a x is called a logarithmic function with base a. The most widely used base for logarithmic functions is the number e, where e = 2.718281828…The function f(x) = log_e x = ln x is called the natural logarithmic function.

Consider how the natural logarithmic function below is used to model human long-term memory performance in the example below.

Example:
Students participating in a psychological experiment attended several lectures on Art History. At the end of the last lecture and every month for the next year, the students were tested to see how much of the material they remembered. The average scores for the group were given by the function f (t) = 75 – 6 ln(t + 1), 0 ≤ t ≤ 12 where t is the time in months and f(t) is the average score at the end of time t. Answer the following questions:
(i) What was the average score on the original (t = 0) exam?
(ii) What was the average score at the end of t = 2 months?
(iii) What was the average score at the end of t = 6 months?
(iv) How long would it take the average score to decrease to 60?

Give an algebraic solution.

Answer

Exercise:
Students in a psychology class were given an exam and then tested monthly with an equivalent exam. The average score for the class was given by the model

f(t) = 80 – 17 log ( t + 1), 0 ≤ t ≤ 12 where t is the time in months.


(i) What was the average score on the original exam?
(ii) What was the average score after 4 months?
(iii) What was the average score after 10 months?
Give an algebraic solution.

Answer