Problem 66 - Applied Calculus

Question

BEHAVIORAL SCIENCE: Response Rate A subject in a psychology experiment gives responses at the rate of \(t / \sqrt{t+1}\) correct answers per minute after \(t\) minutes. a. Find the total number of correct responses from time \(t=0\) to time \(t=15\). b. Verify your answer to part (a) using a graphing calculator.

Step-by-Step Solution

Answer

The total number of correct responses is 28.

Step 1: Understand the Function

The function given is the rate of correct answers per minute: \( r(t) = \frac{t}{\sqrt{t+1}} \). We want to find the total number of correct responses from \( t = 0 \) to \( t = 15 \). This involves finding the integral of the function over the given interval.

Step 2: Set Up the Integral

To find the total number of correct responses, set up the integral of \( r(t) = \frac{t}{\sqrt{t+1}} \) from \( t = 0 \) to \( t = 15 \). The integral is denoted as: \[ \int_{0}^{15} \frac{t}{\sqrt{t+1}} \, dt \]

Step 3: Use Substitution for Integration

Apply the substitution method to the integral. Let \( u = t + 1 \), then \( du = dt \) and \( t = u - 1 \). The limits of integration change accordingly: when \( t = 0, u = 1 \) and when \( t = 15, u = 16 \). Substitute these into the integral:\[ \int_{1}^{16} \frac{u-1}{\sqrt{u}} \, du \]

Step 4: Simplify the Integral

Split the integral: \[ \int_{1}^{16} \frac{u-1}{\sqrt{u}} \, du = \int_{1}^{16} u^{1/2} \, du - \int_{1}^{16} u^{-1/2} \, du \]

Step 5: Integrate Each Part

Calculate each integral separately:1. \( \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C \)2. \( \int u^{-1/2} \, du = 2u^{1/2} + C \)By applying limits, we get:\[ \left[ \frac{2}{3} u^{3/2} \right]_1^{16} - \left[ 2u^{1/2} \right]_1^{16} \]

Step 6: Evaluate the Definite Integral

Calculate each evaluated part using the limits:- \( \left[ \frac{2}{3} \cdot 16^{3/2} \right] - \left[ \frac{2}{3} \cdot 1^{3/2} \right] = \left[ \frac{2}{3} \cdot 64 \right] - \left[ \frac{2}{3} \cdot 1 \right] = \frac{128}{3} - \frac{2}{3} = \frac{126}{3} = 42 \)- \( \left[ 2 \cdot 16^{1/2} \right] - \left[ 2 \cdot 1^{1/2} \right] = [8 \cdot 2] - [2] = 16 - 2 = 14 \) Now subtract both results: \( 42 - 14 = 28 \)

Step 7: Verify with Graphing Calculator

Use a graphing calculator to numerically integrate \( \frac{t}{\sqrt{t+1}} \) from \( t = 0 \) to \( t = 15 \). Confirm that the output matches the analytical solution, which should be 28.

Key Concepts

Psychology Experiments

Psychology experiments are structured studies designed to explore behavioral patterns, cognitive processes, or emotional responses in individuals or groups. In these experiments, researchers seek to understand how various stimuli or conditions affect mental processes or behaviors.

Experiments might involve tasks that assess memory, perception, or reaction times. For instance, a study could explore how the presentation of flashing lights influences a participant's reaction time.

Key components of psychology experiments include:

• **Participants**: The subjects who take part in the study and whose behaviors are analyzed.
• **Variables**: Factors that the experimenter manipulates (independent variables) or measures (dependent variables).
• **Design**: The structure of the experiment, which might be "within-subjects" or "between-subjects" depending on whether the same or different individuals experience the experimental conditions.

Understanding how various conditions affect behavior and thought processes can provide insights into human psychology and lead to the development of theories and applications in areas like clinical psychology and education.

Response Rate in Psychology

Response rate in psychology refers to the speed or quantity of responses to a particular stimulus or task. This is especially relevant in experiments where the participants' reaction or feedback is being measured over time.

In the context of our exercise, the response rate is given by the formula \(\frac{t}{\sqrt{t+1}}\), describing how quickly a subject generates correct answers in relation to time.

Factors influencing response rate include:

• **Task Complexity**: More challenging tasks can lead to slower response rates.
• **Concentration and Fatigue**: As time progresses, a participant's ability to maintain focus can drop, affecting their response rate.
• **Learning and Adaptation**: An individual might get better at the task over time, showing improved response rates as they learn.

Understanding response rates helps researchers make inferences about cognitive load and the effect of experimental conditions on mental processes.

Definite Integral

In calculus, a definite integral is used to calculate the total accumulation of a quantity, which in our case is the total number of correct answers given by a participant over a set period.

The definite integral is denoted as \( \int_{a}^{b} f(x) \, dx \), where \(a\) and \(b\) are the interval limits, and \(f(x)\) is the function being integrated.

Key aspects to remember about definite integrals are:

• **Limits of Integration**: These define the range over which the integration is performed.
• **Result**: The result is a specific numerical value rather than a function.
• **Interpretation**: It provides the "net area" under the curve of the function between the two points.

In our problem, integrating the rate function from \(t=0\) to \(t=15\) helps us find the total number of correct responses provided by the subject.

Substitution Method in Integration

The substitution method in integration is a powerful technique used to simplify the integration process. It involves changing variables to make a complex integral easier to solve.

Here's how substitution works in a nutshell:

• **Identify**: Start by identifying a substitution that simplifies the integral. Typically, this involves setting \( u = g(x) \) where \( g(x) \) is part of the integrand.
• **Change of Variables**: Substitute \( u \) and its derivative \( du \) into the integral to simplify it.
• **Adjust Limits**: When dealing with definite integrals, change the limits of integration to match the new variable \( u \).

In our exercise, the substitution \( u = t + 1 \) simplifies the function, making it more manageable to integrate. This illustrates how substitution can transform a challenging problem into one that is much easier to solve.