Differentiation
Overview
Differentiation is the mathematics of slopes and change. It gives you the gradient of a curve at any point, not just a straight line, and that single idea lets you find where a curve is steepest, where it turns, and where a quantity is at its maximum or minimum.
In this lesson (Extended tier) you will estimate gradients by drawing tangents, differentiate functions of the form a x to the power n, and use the derivative to find gradients and stationary (turning) points, telling maxima from minima.
Objectives
- Estimate gradients of curves by drawing tangents.
- Use the derivatives of functions of the form axn, where a is a rational constant and n is a positive integer or zero, and simple sums of not more than three of these.
- Apply differentiation to gradients and stationary points (turning points).
- Discriminate between maxima and minima by any method.
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Flashcards
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Differentiation. 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.
- Differentiate y = x^2. A. 2x B. x C. 2x^2 D. 2 Answer: A
- For y = x^2, what is the gradient at x = 2? A. 2 B. 4 C. 8 D. 1 Answer: B
- Differentiate y = 3x^2. A. 3x B. 6x C. 6x^2 D. 9x Answer: B
- At a turning point, the gradient dy/dx is: A. 1 B. negative C. zero D. undefined Answer: C
- For y = x^2 - 4x + 1, at what value of x is the turning point? A. x = 1 B. x = 2 C. x = 4 D. x = -2 Answer: B
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