微积分：早期超越函数（第7版）

This lesson introduces functions as rules that assign each input in a domain to exactly one output in a range, emphasizing the importance of the vertical line test and algebraic evaluation. Students learn to determine domains by identifying mathematical constraints and explore how functions model real-world relationships through examples like geometric calculations and difference quotients.

This lesson introduces the concept of limits as a way to describe the behavior of a function as it approaches a specific point, rather than its value at that point. Students learn to evaluate one-sided limits, determine the existence of two-sided limits, and identify vertical asymptotes caused by unbounded growth.

This lesson introduces the Power Rule and the linearity of the derivative, providing an efficient, algorithmic method for differentiating polynomials and power functions. Students learn to apply these rules alongside the Constant and Sum rules to solve for slopes and tangent lines, transitioning from fundamental limit definitions to operational mastery.

This lesson explores the distinction between local and absolute extrema, emphasizing the Extreme Value Theorem's guarantee that continuous functions on closed intervals must attain both maximum and minimum values. Students learn to identify these points by evaluating critical numbers and function endpoints, a process essential for optimizing real-world systems.

This lesson introduces the concept of integration by using Riemann sums to approximate the area under a curve and total displacement through the accumulation of infinitely small intervals. Students learn to refine these estimates using limits and sigma notation, bridging the gap between discrete geometric approximations and the formal definition of the definite integral.

This lesson explores how to calculate the area of regions trapped between two curves by integrating the difference between the upper and lower functions. Students learn to determine integration bounds by finding intersection points and discover how to simplify complex problems by integrating with respect to the y-axis.

This lesson focuses on mastering the "Table of 20" standard integral forms and emphasizes the importance of algebraic or trigonometric simplification before attempting complex integration. It also highlights the necessity of human oversight when using Computer Algebra Systems to ensure mathematical rigor regarding constants of integration and domain-specific absolute values.

This lesson explores integration as a powerful tool for accumulating varying quantities, moving beyond simple area and volume to solve complex problems in physics, economics, and geometry. Students learn to apply the Riemannian logic of partitioning, approximating, and taking limits to calculate arc lengths, hydrostatic forces, and producer surplus.

This lesson introduces differential equations as mathematical models that describe how systems evolve by relating an unknown function to its rate of change. Students learn to identify the order of equations, verify solutions, and use initial conditions to determine unique trajectories within a family of possible curves.

This lesson introduces parametric equations as a way to define curves by expressing $x$ and $y$ as functions of a third variable, $t$. Students will learn how this approach captures the timing, direction, and orientation of motion, allowing for the representation of complex paths that static Cartesian equations cannot describe.

This lesson introduces the formal definition of sequences and their convergence, utilizing L'Hospital's Rule to evaluate limits by bridging discrete sequences with continuous functions. It further explores the rigorous foundations of convergence by examining how monotonicity and boundedness guarantee the existence of a limit.

This lesson introduces the three-dimensional coordinate system ($\mathbb{R}^3$), teaching students how to locate points using ordered triples and identify surfaces defined by coordinate planes and equations. It emphasizes the importance of understanding coordinate environments, as the geometric interpretation of an equation can shift significantly when moving from two to three dimensions.

This lesson introduces vector-valued functions as a way to describe the trajectory of a moving particle through 3D space using parametric equations. Students learn to analyze space curves, such as helices and twisted cubics, by examining their component functions, domains, and geometric properties.

This lesson explores multivariable calculus by using level curves and level surfaces to visualize functions of two and three variables. It also covers key analytical tools, including partial derivatives, gradients, linear approximations, and the use of Lagrange multipliers for constrained optimization.

This lesson introduces double integrals as a method for calculating the volume under a surface by extending the concept of Riemann sums into three dimensions. Students learn to approximate volumes using rectangular partitioning and explore the average value theorem to determine the mean height of a surface over a given region.

This lesson introduces vector fields as functions that assign a vector to every point in space, providing a mathematical framework for modeling physical phenomena like fluid flow, gravity, and electricity. Students will learn to distinguish between scalar and vector fields, calculate gradient fields, and understand the geometric relationship where gradient vectors are perpendicular to level curves.

This lesson introduces second-order linear differential equations, focusing on their standard form, the distinction between homogeneous and nonhomogeneous equations, and the Principle of Superposition. Students learn to find general solutions using linearly independent functions and the auxiliary equation method for constant-coefficient problems.