Use factoring, fractions, exponents, radicals, and equation-solving without getting stuck. These moves make later calculus steps shorter and safer.

Work with sine, cosine, tangent, exponentials, and logarithms as common patterns of motion, growth, decay, and cycles. You will connect their graphs, formulas, and everyday meanings.

Compare average speed, steepness, total distance, area, and accumulated change before formal calculus notation appears. This gives the main questions of calculus a concrete shape.

Trace how problems about motion, astronomy, area, and prediction led to the derivative and the integral. The chapter connects Newton, Leibniz, later notation, rigor, and the tools used in classrooms and software today.

Use limits to describe what a function approaches, even when direct substitution fails. You will estimate limits from graphs and tables, then calculate them with algebra.

Recognize when a graph has no breaks, jumps, holes, or blow-ups, and why that matters. Continuity becomes the bridge between intuitive graphs and reliable calculation.

Use precise limit definitions to prove simple facts about limits, continuity, and slopes. You will practice writing short, honest arguments instead of only copying procedures.

Build the derivative from secant slopes, tangent lines, and instantaneous rate of change. You will compute simple derivatives from the definition and interpret their units.

Use constant, power, sum, product, and quotient rules to take derivatives efficiently. The chapter keeps each rule tied to graphs, rates, and formulas rather than memorization alone.

Handle functions inside other functions with the chain rule. You will apply it to motion, growth, temperature, finance, and other layered relationships.

Differentiate sine, cosine, tangent, exponential, and logarithmic functions. These tools cover waves, rotations, compounding, decay, and many natural growth patterns.

Find slopes when variables are mixed together or when one variable is defined through another. You will use implicit differentiation and inverse-function reasoning without needing to solve every equation first.

Use derivatives to build tangent-line estimates and small-change formulas. This chapter shows why local linear models are often the first useful approximation.

Find maximum and minimum values with derivatives, endpoints, and constraints. You will solve practical design, cost, distance, and efficiency problems from start to finish.

Connect several changing quantities in one situation, such as ladders sliding, shadows moving, or tanks filling. The focus is setting up the relationship before differentiating.

Use first and second derivatives to read increasing, decreasing, concavity, inflection points, and asymptotes. You will turn derivative information into accurate sketches and decisions.

Reverse differentiation to find antiderivatives and families of functions. You will see constants of integration as missing starting information, not as decoration.

Build the definite integral from sums of many small pieces. You will connect area, signed area, distance, mass, and total change under one idea.

Use the Fundamental Theorem of Calculus to connect rates and totals. This chapter shows why derivatives and integrals are two sides of the same process.

Use substitution to integrate functions created by the chain rule. You will practice choosing the inside quantity and changing limits cleanly.

Use integration by parts for products such as polynomial times exponential, logarithmic, or trigonometric functions. The chapter also covers choosing which part to differentiate and which to integrate.

Integrate rational, trigonometric, and root-heavy expressions with identities, partial fractions, and targeted substitutions. You will build judgment about which technique fits which form.

Approximate integrals with rectangles, trapezoids, Simpson’s rule, and simple error checks. This prepares you for cases where no neat antiderivative exists.

Use integrals to compute area between curves, volumes, arc length, work, pressure, centers of mass, and average value. The emphasis is choosing the small slice that represents the situation.

Handle infinite intervals, vertical asymptotes, and quantities that may or may not add up to a finite total. You will decide when an improper integral converges.

Model exponential growth, decay, cooling, mixing, and population change with differential equations. You will solve basic initial-value problems and check answers against the situation.

Use direction fields, separable equations, and first-order linear equations to predict behavior over time. The chapter balances exact solutions with qualitative reasoning.

Work with sequences as ordered lists that may settle down, grow, oscillate, or diverge. This prepares the ground for infinite series.

Decide whether infinite sums converge using geometric series, p-series, comparison tests, ratio tests, root tests, and alternating-series ideas. You will connect convergence tests to the shape of the terms.

Represent functions with power series and Taylor polynomials. You will approximate sine, cosine, exponential, logarithmic, and other functions while tracking error.

Use parametric equations and polar coordinates for curves that are awkward in ordinary x-y form. You will compute slopes, areas, and lengths for paths and circular patterns.

Work with vectors, dot products, cross products, lines, planes, and distances in three-dimensional space. These tools make multivariable calculus possible.

Study functions with more than one input using level curves, partial derivatives, gradients, and tangent planes. You will interpret how a surface changes in different directions.

Find local extrema, constrained extrema, and saddle points for multivariable functions. Lagrange multipliers become a practical way to optimize under a rule.

Use double and triple integrals to add up quantities over regions, solids, and variable densities. You will change order of integration and use rectangular, polar, cylindrical, and spherical coordinates.

Work with vector fields, line integrals, circulation, and flux. These ideas describe forces along paths, fluid flow, electric fields, and other directional quantities.

Use Green’s theorem, Stokes’ theorem, and the divergence theorem to connect local behavior with boundary behavior. The chapter shows one grand pattern behind several major results.

Use graphing tools, spreadsheets, computer algebra systems, and Python-style notebooks to calculate, visualize, and check calculus work. You will also see where software can mislead you.

See how automatic differentiation computes exact derivatives through code, powering optimization in machine learning, simulation, and scientific computing. You will compare it with symbolic differentiation and numerical finite differences.

Follow a complete calculus modeling workflow: state the question, choose variables, build assumptions, calculate, check units, test sensitivity, and explain the result. The project ties derivatives, integrals, numerical work, and communication together.

Connect calculus to next steps in physics, engineering, economics, biology, data science, optimization, and real analysis. You will identify proof-of-skill projects, study paths, useful software habits, and ways to keep your skills current.