Chapter 1: Real Numbers
* Defines real numbers as the set of rational and irrational numbers.
* Examines properties of real numbers, including ordering, addition, and multiplication.
* Real-world example: Calculating the total cost of groceries by multiplying the quantity of each item by its price and adding the results.
Chapter 2: Sequences
* Introduces sequences as ordered lists of numbers.
* Explores limits of sequences, convergence, and divergence.
* Real-world example: Modeling the spread of a virus in a population using a sequence of population sizes.
Chapter 3: Functions
* Defines functions as mappings between sets.
* Studies properties of functions, such as continuity, differentiability, and integrability.
* Real-world example: Describing the relationship between the height of a projectile and its time in the air using a quadratic function.
Chapter 4: Limits and Continuity
* Examines the concept of a limit and its relationship to continuity.
* Investigates different types of discontinuity and their implications.
* Real-world example: Determining the behavior of a weather forecast model as the time interval approaches zero.
Chapter 5: Differentiation
* Introduces the derivative as the instantaneous rate of change of a function.
* Examines rules for differentiation and applications to finding critical points, maxima, and minima.
* Real-world example: Using the derivative to determine the optimal production level to maximize profit.
Chapter 6: Applications of the Derivative
* Explores applications of differentiation in optimization, curve sketching, and related rates problems.
* Real-world example: Designing a bridge arch with a shape that minimizes wind resistance.
Chapter 7: Integration
* Defines the integral as the area under a curve.
* Investigates properties of integrals, including the Fundamental Theorem of Calculus.
* Real-world example: Calculating the volume of a liquid in a tank using integration.
Chapter 8: Applications of Integration
* Explores applications of integration in finding volumes, surface areas, centers of mass, and work.
* Real-world example: Determining the total force acting on a dam by integrating the pressure across its surface.
Chapter 9: Sequences and Series
* Examines sequences and infinite series and their convergence.
* Introduces tests for convergence, including the Ratio Test and the Integral Test.
* Real-world example: Determining if a series representing the probability of success in a game converges.
Chapter 10: Power Series
* Introduces power series and their convergence.
* Studies applications of power series in representing functions and solving differential equations.
* Real-world example: Using a power series to approximate the value of pi.
