Rates Of Change And Small Increments
Overview
Differentiation measures how one quantity changes as another changes. But often two or more quantities change together over time, and we want to link their rates: if a balloon's radius grows, how fast does its volume grow? Connected rates of change answer exactly these questions using the chain rule.
In this lesson you will use the chain rule to connect rates, with \(\frac{dy}{dt}=\frac{dy}{dx}\times\frac{dx}{dt}\), and learn how a derivative estimates a small change through \(\delta y \approx \frac{dy}{dx}\,\delta x\). Together they show how calculus handles change that ripples from one quantity to another.
Objectives
- Apply differentiation to connected rates of change, small increments and approximations.
Lesson Note
Available on the Green Bridge App
Download the Green Bridge CBT app on your phone or computer to access full lesson notes, practice questions, and more.
Lesson Evaluation
Congratulations on completing the lesson on Rates Of Change And Small Increments. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- The chain rule for connected rates states that dy/dt equals: A. dy/dx + dx/dt B. dy/dx x dx/dt C. dx/dt / dy/dx D. dy/dx - dx/dt Answer: B
- The area of a square is A = x^2. If x increases at 0.5 cm/s, what is dA/dt when x = 4? A. 2 cm^2/s B. 4 cm^2/s C. 8 cm^2/s D. 16 cm^2/s Answer: B
- The small-increment approximation is: A. delta y = dy/dx + delta x B. delta y ~ (dy/dx) delta x C. delta y ~ (dx/dy) delta x D. delta y = delta x / (dy/dx) Answer: B
- For y = x^3, estimate the change in y when x increases from 2 by delta x = 0.01. A. 0.03 B. 0.06 C. 0.12 D. 1.2 Answer: C
- In connected rates, dy/dx is found by: A. Integrating the relationship B. Differentiating the relationship between the quantities C. Dividing the two rates D. Adding the two rates Answer: B
Available on the Green Bridge App
Download the Green Bridge CBT app on your phone or computer to access full lesson notes, practice questions, and more.
Available on the Green Bridge App
Download the Green Bridge CBT app on your phone or computer to access full lesson notes, practice questions, and more.
Recommended Books
Practice Mock Questions
Want to practice mock questions on Rates Of Change And Small Increments? Download the Green Bridge CBT app to access mock questions and full practice assessments for this topic.
Download The App On Google Playstore
Everything you need to excel in JAMB, WAEC & NECO
