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Chapter Analysis
Intermediate13 pages • EnglishQuick Summary
This chapter introduces the concept of limits and derivatives, essential tools in calculus. It explains the mathematical process for finding limits and introduces the derivative as a measure of how a function changes at any given point. The chapter covers the derivation of standard functions, employs rules like the product rule and quotient rule, and discusses their applications. There are numerous examples and exercises to help the student practice.
Key Topics
- •Introduction to limits
- •Understanding derivatives
- •The product rule
- •The quotient rule
- •Applications of derivatives
- •Examples and exercises
Learning Objectives
- ✓Understand the basic concept of limits and how to calculate them.
- ✓Learn to find derivatives using different mathematical rules.
- ✓Apply derivatives to solve real-world problems.
- ✓Comprehend the graphical interpretation of derivatives as slopes.
- ✓Analyze the behavior of functions using derivatives.
Questions in Chapter
Find the derivative of the following functions from first principle: x−, 1/(x), sin(x + 1), cos(x – π/8)
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Find the derivative of x^2 – 2 at x = 10.
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Prove that f'(1) = f'(100) = 0 for f(x) = 100x + 99x + 2...
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Find the derivative of cos(x) from first principle.
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Find the derivative of the function (x + a)(x + b) where a and b are constants.
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Additional Practice Questions
Evaluate the derivative of f(x) = x^3 at x = 2 using the product rule.
mediumAnswer: f'(x) = 3x^2, hence f'(2) = 12.
What is the derivative of g(x) = e^x using first principles?
hardAnswer: The derivative g'(x) = e^x because the limit as h approaches 0 of [(e^(x+h) - e^x)/h] results in e^x.
Compute the limit of the function f(x) = (x^2 - 4x)/(x-4) as x approaches 4.
easyAnswer: The limit is 0, after factorizing and canceling the common terms, applying the limit gives f(x) = x as x approaches 4, resulting in 0.
Find the derivative of the function h(x) = ln(x) using the quotient rule.
mediumAnswer: h'(x) = 1/x, derived through recognizing ln(x) as ln(x) * 1 and applying the quotient rule.
Explain the significance of the derivative in understanding the rate of change.
easyAnswer: The derivative provides the instantaneous rate of change of a function, representing velocity in physical problems.