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Chapter Analysis
Intermediate31 pages • EnglishQuick Summary
The chapter on 'Application of Derivatives' explores various practical uses of derivatives in mathematics and daily life. It covers the rate of change of quantities, determining the equations of tangents and normals to curves, and finding turning points to identify local maxima and minima. The chapter also delves into using derivatives for identifying intervals of increase and decrease of functions and approximating values. Students will engage with numerous examples and exercises to bolster their understanding.
Key Topics
- •Rate of change of quantities
- •Equations of tangent and normal to curves
- •Turning points and local maxima/minima
- •Increasing and decreasing intervals
- •Applications in economics and engineering
- •Approximation using derivatives
- •Marginal cost and revenue
- •Optimization problems
Learning Objectives
- ✓Understand and calculate the rate of change of a quantity.
- ✓Determine the equations of the tangent and normal at a point on a curve.
- ✓Use derivatives to find and classify turning points of a function.
- ✓Identify intervals where functions increase or decrease.
- ✓Apply derivatives in real-world optimization problems.
- ✓Use derivatives for estimating and approximating values.
Questions in Chapter
Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.
Answer: The rate of change is 6π cm/s.
Page 148
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Answer: The enclosed area is increasing at the rate of 80π cm²/s.
Page 151
Additional Practice Questions
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
mediumAnswer: The volume of the cube is increasing at the rate of 900 cm³/s.
A circular balloon is being inflated such that its radius increases at a rate of 4 cm/s. What is the rate of change of its volume when the radius is 15 cm?
hardAnswer: The rate of change of the volume is 12,000π cm³/s.
Find the local maximum and minimum values of the function f(x) = x³ - 3x + 2.
mediumAnswer: The local maximum is 4 at x = -1, and the local minimum is -2 at x = 1.
Determine the intervals on which the function f(x) = x² - 6x + 9 is decreasing.
easyAnswer: The function is decreasing on (-∞, 3) and increasing on (3, ∞).
Prove that the function f(x) = e^(2x) is an increasing function on R.
mediumAnswer: Since f'(x) = 2e^(2x) is positive for all x in R, f is an increasing function.