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Chapter Analysis
Advanced40 pages • EnglishQuick Summary
Chapter 5 of the Class 12 mathematics textbook covers the concepts of continuity and differentiability. It starts by defining and explaining the concept of a continuous function and how continuity is analyzed at different points in the function's domain. The chapter further discusses the differentiation of functions, including the application of derivatives to determine continuity and introduces the derivatives of composite, implicit, and inverse trigonometric functions. The chapter concludes with exercises to solidify the understanding of these key mathematical principles.
Key Topics
- •Continuity of functions
- •Differentiability and its implications
- •Derivatives of composite functions
- •Derivatives of implicit functions
- •Inverse trigonometric functions
- •Parametric functions
- •Applications of derivatives
- •Algebra of continuous functions
Learning Objectives
- ✓Understand the formal definition of continuity and determine if a function is continuous at a given point.
- ✓Apply the concept of limits to analyze function behavior.
- ✓Distinguish between differentiability and continuity for various functions.
- ✓Use the chain rule to differentiate composite functions efficiently.
- ✓Analyze the behavior of parametric and inverse trigonometric functions using differentiation.
- ✓Apply derivatives for practical problem-solving and mathematical exploration.
Questions in Chapter
Is the function defined by f(x) = x² - sin x + 5 continuous at x = π?
Page 117
Discuss the continuity of the following functions: (a) f(x) = sin x + cos x (b) f(x) = sin x – cos x.
Page 117
Examine the continuity of f, where f is defined by sin x cos x, if 0 < x < 1.
Page 118
Discuss the continuity of the function f defined by 1, if x < 0; 0, if x = 0; 1, if x > 0.
Page 118
Differentiate w.r.t. x the function sin⁻¹(x³), 0 ≤ x ≤ 1.
Page 145
Additional Practice Questions
What conditions must be met for a function to be differentiable at a point?
mediumAnswer: A function is differentiable at a point if the function is continuous at that point and the left-hand derivative equals the right-hand derivative at that point.
How would you prove that |x| is not differentiable at x = 0?
hardAnswer: To prove |x| is not differentiable at x = 0, we show that the left-hand derivative limit and the right-hand derivative limit at x = 0 are not equal.
Explain the geometric interpretation of the derivative at a point.
easyAnswer: The derivative of a function at a point is the slope of the tangent to the function's graph at that point.
How can the chain rule be applied to differentiate composite functions?
mediumAnswer: The chain rule states that if a function y can be expressed as a composite of two functions u and v, that is, y = v(u(x)), then dy/dx = (dv/du) * (du/dx).
What is the importance of understanding continuity in the context of calculus?
easyAnswer: Understanding continuity is crucial for calculus as it ensures that functions behave predictably and allows the application of derivatives and integrals.