Chapter 1: Limits and Continuity
* Concept: Introduces the concept of a limit, which describes the value a function approaches as the input approaches a particular point.
* Example: The function f(x) = x^2 approaches 4 as x approaches 2, so the limit of f(x) as x approaches 2 is 4.
Chapter 2: Derivatives
* Concept: Defines the derivative of a function as the instantaneous rate of change of the function with respect to its input.
* Example: The derivative of f(x) = x^3 is f'(x) = 3x^2, which represents the rate of change of f(x) at any given point x.
Chapter 3: Applications of Derivatives
* Concept: Explores various applications of derivatives, including finding critical points, extrema, and rates of change.
* Example: To find the minimum of the function f(x) = x^2 + 2x, take the derivative f'(x) = 2x + 2 and set it equal to 0. Solving for x gives x = -1, which is the minimum point.
Chapter 4: Integrals
* Concept: Introduces the integral as the operation that reverses differentiation. The integral of a function represents the area under the curve of the function.
* Example: The integral of f(x) = x^2 over the interval [0, 2] is (2^3 - 0^3) / 3 = 8/3, which represents the area under the curve of f(x) from x = 0 to x = 2.
Chapter 5: Applications of Integrals
* Concept: Explores applications of integrals, including finding volumes, areas, and lengths of curves.
* Example: To find the volume of the solid generated by rotating the function f(x) = x^2 about the x-axis over the interval [0, 1], use the formula V = π∫[0,1] f(x)^2 dx = π∫[0,1] x^4 dx = π(1/5) = π/5.
Chapter 6: Series
* Concept: Introduces the concept of series, which are sums of infinite sequences of numbers.
* Example: The series 1 + 1/2 + 1/4 + 1/8 + ... is an infinite geometric series with a common ratio of 1/2. Using the formula for the sum of an infinite geometric series, it can be shown that the sum of this series is 2.
Chapter 7: Vectors and Functions of Several Variables
* Concept: Extends the concepts of calculus to functions of several variables and introduces vectors.
* Example: The function f(x, y) = x^2 + y^2 represents a surface in three-dimensional space. The gradient of f(x, y) is a vector that points in the direction of the greatest rate of change of f(x, y), which is given by ∇f = (2x, 2y).
