Problem 67 - Calculus an Applied Approach

Question

Psychology Early in the twentieth century, an intelligence test called the Stanford-Binet Test (more commonly known as the IQ test) was developed. In this test, an individual's mental age \(M\) is divided by the individual's chronological age \(C\) and the quotient is multiplied by \(100 .\) The result is the individual's \(I Q .\) $$I Q(M, C)=\frac{M}{C} \times 100$$ Find the partial derivatives of \(I Q\) with respect to \(M\) and with respect to \(C .\) Evaluate the partial derivatives at the point \((12,10)\) and interpret the result. (Source: Adapted from Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth Edition)

Step-by-Step Solution

Answer

The partial derivative of IQ with respect to M is 10, and with respect to C is -12 at the point (12,10). This suggest that an increase in mental maturity increases an individual's IQ score, while increasing chronological age decreases the IQ score, assuming other variables are kept constant.

Step 1: Compute Partial Derivative with Respect to M

The partial derivative of IQ with respect to M, written as \(\partial IQ / \partial M\), is computed by taking the derivative of the IQ function with respect to M, keeping C constant. The function is in the form of \(u/v\), making use of the quotient rule. So, \(\partial IQ / \partial M = 100 / C\)

Step 2: Compute Partial Derivative with Respect to C

The partial derivative of IQ with respect to C, \(\partial IQ / \partial C\), is computed by taking the derivative of the IQ function with respect to C, keeping M constant. Still the quotient rule is used, yielding \(\partial IQ / \partial C = -100M / C^2\)

Step 3: Evaluate the Partial Derivatives at (12, 10)

Substitute \(M = 12\) and \(C = 10\) into the expressions derived in steps 1 and 2. So, \(\partial IQ / \partial M\)|_(12,10) = 100/10 = 10.\n\nAnd \(\partial IQ / \partial C\)|_(12,10) = -100(12)/10^2 = -12.

Step 4: Interpret the Result

The partial derivative of IQ with respect to M, evaluated at (12, 10), shows how IQ changes with a small change in M. In this case, an additional year of mental maturity results in an increase of 10 IQ points. The partial derivative of IQ with respect to C, evaluated at (12, 10), is how IQ changes with a one-year increase in chronological age. Here, an additional year of actual age results in a decrease of 12 IQ points, assuming mental age remains constant.

Key Concepts

Stanford-Binet Test

The Stanford-Binet Test is a foundational assessment tool in psychology designed to measure human intelligence. Developed in the early 20th century, the test evaluates cognitive abilities through a variety of questions and tasks. The score that results from this test is popularly known as an Intelligence Quotient, or IQ. To obtain an IQ score, examiners assess a subject's mental age (M) — the age level at which they perform cognitively — and compare it to their chronological age (C), which is their actual age.

The test has evolved over time to suit different age ranges and to accommodate updated theories of intelligence. As such, professionals widely use the Stanford-Binet Test in educational fields and psychological evaluation to identify cognitive strengths and weaknesses in both children and adults.

IQ Test Calculation

Calculating one's IQ score from the Stanford-Binet Test hinges on a simple formula: \(IQ(M, C) = \frac{M}{C} \times 100\). By dividing a person’s mental age (M) by their chronological age (C) and multiplying the result by 100, we get the IQ score.

A child with a mental age that matches their chronological age will have an IQ of 100, which is considered the average score. For instance, if a 10-year-old child has a mental age of 10, the child's IQ would be calculated as \(\frac{10}{10} \times 100 = 100\). Alterations in this ratio indicate variations in intellectual development relative to age peers. For example, a child who is 10 years old but has the mental age of a 12-year-old would have an IQ of 120, illustrating advanced cognitive abilities.

Quotient Rule

In calculus, the quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. The rule states that if you have a function \( g(x) \) that is the ratio of two functions, \( u(x) \) and \( v(x) \), in the form \( g(x) = \frac{u(x)}{v(x)} \), then the derivative of \( g(x) \), denoted as \( g'(x) \), is given by:
\[ g'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{v(x)^2} \]
In the case of the IQ calculation, we applied the quotient rule to find how changing either the mental age (M) or the chronological age (C) affects the IQ score.

Mental Age versus Chronological Age

Understanding the distinction between mental age and chronological age is pivotal in interpreting IQ test results. Mental age refers to the intellectual development level as determined by the Stanford-Binet Test (or similar tests), while chronological age is the actual time a person has lived, measured in years. The comparison of the two can provide insight into an individual's cognitive abilities. If a person’s mental age outpaces their chronological age, it suggests higher intellectual potential relative to their age bracket. Conversely, a lower mental age compared to chronological age can indicate developmental delays.

IQ scores draw upon this comparison, making it a powerful tool in educational planning. A child identified with an above-average IQ might benefit from advanced learning programs, while one with a below-average IQ might receive support tailored to their learning needs.