Suppose That The Number Of Facts Of A Certain Type That A Person Can Remember After $t$ Hours Is Given By:$f(t) = \frac{90t}{99t - 90}$Find The Rate At Which The Number Of Facts Remembered Is Changing After 2 Hours.Answer:

Feb 26, 2025 by ADMIN 223 views

Derivatives and Rates of Change: A Mathematical Analysis

In various fields, such as science, engineering, and economics, understanding rates of change is crucial for making informed decisions and predicting outcomes. One of the fundamental concepts in calculus is the derivative, which measures the rate at which a function changes as its input changes. In this article, we will explore how to find the rate at which the number of facts remembered is changing after a certain period, given a specific function that models this phenomenon.

The Function: Modeling the Number of Facts Remembered

The function that models the number of facts of a certain type that a person can remember after $t$ hours is given by:

f(t) = \frac{90t}{99t - 90}

This function takes the time $t$ as input and returns the number of facts remembered as output. To find the rate at which the number of facts remembered is changing, we need to find the derivative of this function with respect to time.

Finding the Derivative

To find the derivative of the function $f(t)$ , we will use the quotient rule, which states that if $f(t) = \frac{g(t)}{h(t)}$ , then $f'(t) = \frac{h(t)g'(t) - g(t)h'(t)}{(h(t))^2}$ . In this case, $g(t) = 90t$ and $h(t) = 99t - 90$ .

First, we need to find the derivatives of $g(t)$ and $h(t)$ :

g'(t) = 90

h'(t) = 99

Now, we can plug these values into the quotient rule formula:

f'(t) = \frac{(99t - 90)(90) - (90t)(99)}{(99t - 90)^2}

Simplifying this expression, we get:

f'(t) = \frac{8910t - 8100 - 8910t}{(99t - 90)^2}

f'(t) = \frac{-8100}{(99t - 90)^2}

Evaluating the Derivative at $t = 2$

Now that we have found the derivative of the function $f(t)$ , we can evaluate it at $t = 2$ to find the rate at which the number of facts remembered is changing after 2 hours.

f'(2) = \frac{-8100}{(99(2) - 90)^2}

f'(2) = \frac{-8100}{(198 - 90)^2}

f'(2) = \frac{-8100}{(108)^2}

f'(2) = \frac{-8100}{11664}

f'(2) = -0.695

Therefore, the rate at which the number of facts remembered is changing after 2 hours is approximately -0.695 facts per hour.

In this article, we used the quotient rule to find the derivative of a function that models the number of facts remembered after a certain period. We then evaluated the derivative at $t = 2$ to find the rate at which the number of facts remembered is changing after 2 hours. This type of analysis is essential in various fields, such as science, engineering, and economics, where understanding rates of change is crucial for making informed decisions and predicting outcomes.

The concept of derivatives and rates of change is a fundamental aspect of calculus. In this article, we used the quotient rule to find the derivative of a function that models the number of facts remembered after a certain period. The derivative of a function represents the rate at which the function changes as its input changes. In this case, the derivative of the function $f(t)$ represents the rate at which the number of facts remembered is changing after a certain period.

The rate at which the number of facts remembered is changing after 2 hours is approximately -0.695 facts per hour. This means that after 2 hours, the number of facts remembered is decreasing at a rate of approximately 0.695 facts per hour.

The concept of derivatives and rates of change has numerous applications in various fields, such as:

Science: Understanding rates of change is crucial in scientific research, where scientists need to analyze data and make predictions about future outcomes.
Engineering: Engineers use derivatives and rates of change to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Economists use derivatives and rates of change to analyze economic data and make predictions about future economic trends.

In future work, we can explore other applications of derivatives and rates of change, such as:

Optimization: Using derivatives and rates of change to optimize systems and processes.
Predictive Modeling: Using derivatives and rates of change to build predictive models that can forecast future outcomes.
Data Analysis: Using derivatives and rates of change to analyze data and make informed decisions.

Calculus: A First Course, by Michael Spivak
Calculus: Early Transcendentals, by James Stewart
Derivatives and Rates of Change, by Michael Sullivan
Derivatives and Rates of Change: A Q&A Article

In our previous article, we explored the concept of derivatives and rates of change, and how they can be used to analyze and predict outcomes in various fields. In this article, we will answer some of the most frequently asked questions about derivatives and rates of change.

Q: What is a derivative?

A: A derivative is a measure of how a function changes as its input changes. It represents the rate at which the function changes at a given point.

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a function that is the quotient of two functions. It is given by:

f'(t) = \frac{h(t)g'(t) - g(t)h'(t)}{(h(t))^2}

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you can use the following steps:

Identify the function and its input variable.
Determine the type of function (e.g. polynomial, rational, trigonometric).
Use the appropriate derivative formula (e.g. power rule, quotient rule, product rule).
Simplify the resulting expression.

Q: What is the difference between a derivative and a rate of change?

A: A derivative represents the rate at which a function changes at a given point, while a rate of change represents the rate at which a quantity changes over a given interval.

Q: How do I evaluate a derivative at a specific point?

A: To evaluate a derivative at a specific point, you can substitute the point into the derivative formula and simplify the resulting expression.

Q: What are some common applications of derivatives and rates of change?

A: Some common applications of derivatives and rates of change include:

Optimization: Using derivatives and rates of change to optimize systems and processes.
Predictive Modeling: Using derivatives and rates of change to build predictive models that can forecast future outcomes.
Data Analysis: Using derivatives and rates of change to analyze data and make informed decisions.

Q: What are some common mistakes to avoid when working with derivatives and rates of change?

A: Some common mistakes to avoid when working with derivatives and rates of change include:

Not simplifying the derivative expression: Failing to simplify the derivative expression can lead to incorrect results.
Not evaluating the derivative at the correct point: Evaluating the derivative at the wrong point can lead to incorrect results.
Not considering the context of the problem: Failing to consider the context of the problem can lead to incorrect results.

Q: How can I practice working with derivatives and rates of change?

A: You can practice working with derivatives and rates of change by:

Solving problems: Solving problems that involve derivatives and rates of change can help you develop your skills and build your confidence.
Using online resources: Using online resources, such as calculators and interactive tools, can help you visualize and understand the concepts.
Working with a tutor or mentor: Working with a tutor or mentor can provide you with personalized guidance and support.

In this article, we answered some of the most frequently asked questions about derivatives and rates of change. We hope that this article has provided you with a better understanding of these concepts and how they can be used to analyze and predict outcomes in various fields. If you have any further questions or need additional guidance, please don't hesitate to ask.

Derivatives and rates of change are fundamental concepts in calculus that have numerous applications in various fields. In this article, we explored some of the most frequently asked questions about derivatives and rates of change. We hope that this article has provided you with a better understanding of these concepts and how they can be used to analyze and predict outcomes in various fields.

Derivatives and rates of change have numerous applications in various fields, including:

Science: Understanding derivatives and rates of change is crucial in scientific research, where scientists need to analyze data and make predictions about future outcomes.
Engineering: Engineers use derivatives and rates of change to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Economists use derivatives and rates of change to analyze economic data and make predictions about future economic trends.