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Mathematics · Subject
Calculus I
8 units · 41 lessons · 1 articles · 9 supplements
Unit 1 - Foundations for Change
8 items8L0A
- 01lesson
Functions as Calculus Objects
Treat a function as an input-output assignment whose domain, range, units, and meaning must survive changes of representation.
- 02lesson
Reading Graphs, Tables, and Function Models
Extract global and local behavior from graphs, tables, and formulas while respecting sampling limits, units, and model assumptions.
- 03lesson
Transforming Functions with Purpose
Predict and explain vertical, horizontal, reflective, and scaling transformations from changes made inside and outside a parent function.
- 04lesson
Parameters Across Function Families
Interpret parameters and invariant structure across linear, quadratic, exponential, logarithmic, and trigonometric function families.
- 05lesson
Average Rate of Change and Secant Slope
Interpret average rate of change as an output change per input change and as the slope of a secant line across an interval.
- 06lesson
Difference Quotients as Movable Secants
Build, simplify, and interpret x-plus-h and x-minus-a difference quotients as families of secant slopes.
- 07lesson
Algebraic Restructuring for Calculus
Use factoring, common denominators, and conjugates to reveal local structure while preserving domain restrictions and equivalence.
- 08lesson
Radians, Neighborhoods, and Calculus Readiness
Connect radians, absolute-value neighborhoods, exponent and logarithm laws, and domain discipline to the language calculus uses.
Unit 2 - Limits and Continuity
11 items11L0A
- 01lesson
The Instantaneous-Rate Problem
Discover why ordinary average rates cannot directly measure change at one instant and why a controlled approach is mathematically necessary.
- 02lesson
Secants Approaching a Local Rate
Investigate shrinking secant intervals from both directions and decide whether their slopes support one local-rate destination.
- 03lesson
Reading Limit Notation and Nearby Behavior
Read finite limit notation precisely and separate a function's nearby destination from its assigned value at the target input.
- 04lesson
Evidence and Failure in Finite Limits
Evaluate finite-limit claims using graphs, tables, formulas, counterexamples, and calibrated distinctions among evidence, conjecture, and proof.
- 05lesson
One-Sided Limit Notation and Direction
Read left-hand and right-hand limit notation, connect direction to the input domain, and estimate directional destinations from graphs, tables, and formulas.
- 06lesson
Directional Agreement and Two-Sided Existence
Use matching one-sided limits as the necessary and sufficient criterion for a two-sided limit, including jumps, endpoints, and unbounded behavior.
- 07lesson
Limit Laws and Direct Substitution
Use limit laws and continuity to compute finite limits efficiently while recognizing when direct substitution is justified and when it is only a diagnostic.
- 08lesson
Computing Nontrivial Limits
Resolve indeterminate finite limits by selecting factoring, conjugates, or common denominators to expose valid nearby structure.
- 09lesson
Making Approaches Precise: Epsilon and Delta
Translate finite limit intuition into epsilon-delta control by choosing an input neighborhood that guarantees a requested output tolerance.
- 10lesson
Infinite Limits and Limits at Infinity
Distinguish unbounded behavior near a finite input from long-run behavior as inputs grow without bound, using directional and asymptotic reasoning.
- 11lesson
Continuity and the Intermediate Value Theorem
Test continuity by matching a function's nearby behavior to its assigned value, then use continuous change to prove that intermediate outputs occur.
Unit 3 - Derivatives as Local Behavior
5 items5L0A
- 01lesson
Derivative as Instantaneous Rate and Tangent Slope
Build the derivative from a limiting average rate, connecting the difference quotient to tangent slope and units in physical models.
- 02lesson
The Derivative as a Function
Move from one derivative value to a derivative function, interpreting its formula, graph, table, sign, and domain as local-rate information.
- 03lesson
Basic Derivative Rules
Derive and use the constant, sum, constant-multiple, and power rules while keeping each shortcut connected to the limit definition.
- 04lesson
Product, Quotient, and Chain Rules
Differentiate changing products, quotients, and compositions by identifying their structure and tracking each linked rate of change.
- 05lesson
Implicit Differentiation and Inverse Functions
Find derivatives from relations that are not solved for y, then explain why inverse functions have reciprocal tangent slopes.
Unit 4 - Using Derivatives
7 items7L0A
- 01lesson
Linearization and Differentials
Use a tangent line as a local linear model, estimate small changes with differentials, and state the limits of a derivative-based approximation.
- 02lesson
Higher Derivatives and Motion
Interpret first and second derivatives as linked physical rates, use units to distinguish position, velocity, and acceleration, and read concavity as changing slope.
- 03lesson
Extreme Values and Optimization
Use derivatives to identify and test extrema on a stated domain, then build and interpret a one-variable optimization model.
- 04lesson
Mean Value Theorem and Consequences
Connect average rate of change to an interior instantaneous rate, verify the theorem's hypotheses, and use it to justify monotonicity and uniqueness conclusions.
- 05lesson
Graphing with First and Second Derivatives
Construct a defensible graph sketch from domain, anchors, derivative signs, extrema, and concavity rather than visual guesswork.
- 06lesson
Related Rates
Model simultaneous changes with a constraint equation, differentiate before substituting, and report the resulting rate with correct units.
- 07lesson
Newton's Method and Modeling Error
Use tangent-line iteration to approximate roots, check convergence responsibly, and distinguish numerical error from model limitations.
Unit 5 - Accumulation, Integrals, and the Fundamental Theorem
6 items6L0A
- 01lesson
Antiderivatives and Simple Differential Equations
Reverse differentiation to construct families of functions, use initial conditions to select one member, and interpret simple rate equations with units.
- 02lesson
Area, Sigma Notation, and Riemann Sums
Build definite integration from signed-area approximations, read sigma notation, and interpret Riemann sums as accumulated rate times width.
- 03lesson
Accumulation Functions and Net Change
Treat a variable-bound integral as a function, explain why its derivative recovers the integrand, and use rate integration to compute net change.
- 04lesson
The Fundamental Theorem of Calculus
Connect accumulated signed area to antiderivatives, evaluate definite integrals by endpoints, and preserve the meaning and units of net change.
- 05lesson
Substitution as Reverse Chain Rule
Use u-substitution to reverse a nested derivative pattern, transform differentials and bounds correctly, and verify an antiderivative by the Chain Rule.
- 06lesson
Numerical Integration and Average Value
Approximate definite integrals with midpoint and trapezoidal rules, discuss error responsibly, and interpret average function value as equal-area height.
Unit 6 - Applications of Integration and Course Synthesis
4 items4L0A
- 01lesson
Area Between Curves
Model a bounded planar region with vertical or horizontal slices, determine the correct subtraction order, and split integrals when boundaries switch.
- 02lesson
Volumes by Slicing and Washers
Construct volumes from cross-sectional area, use disks and washers for solids of revolution, and choose slice direction from the rotation axis.
- 03lesson
Work, Motion, and Total Change
Interpret definite integrals as work, displacement, distance, and other accumulated changes while using units and signs to choose the correct quantity.
- 04lesson
Calculus I Synthesis and Calculus II Readiness
Connect functions, limits, derivatives, and integrals into one coherent model-building framework and identify the habits that prepare you for further calculus.
Calculus I Supplements
9 items0L0A9S
- 01supplement
Supplement: Function Notation, Domain, and Composition
Read functions as input-output relationships, determine domains, and follow a quantity through a composition before using the Chain Rule.
- 02supplement
Supplement: Rules of Exponents
Rebuild exponent laws from repeated multiplication, including zero and negative exponents, rational exponents, and the restrictions that make each rule valid.
- 03supplement
Supplement: Logarithm Laws and Exponential Inverses
Use logarithms as inverse functions, apply product, quotient, and power laws with conditions, and recognize transformations that are not valid.
- 04supplement
Supplement: Calculus Notation and How to Read It
Decode function, limit, derivative, and integral notation by connecting each symbol to the quantity and action it represents.
- 05supplement
Supplement: Trigonometric Functions and the Unit Circle
Use radians, the unit circle, and periodic graphs to interpret sine, cosine, tangent, and their signs before applying calculus rules.
- 06supplement
Supplement: Trigonometric Identities
Use unit-circle relationships to verify and apply Pythagorean, reciprocal, quotient, and symmetry identities without treating them as disconnected formulas.
- 07supplement
Supplement: Derivative Rules and Method Choice
Select and apply derivative rules by reading expression structure, including constant, power, product, quotient, chain, exponential, logarithmic, and trigonometric forms.
- 08supplement
Supplement: Rules of Integration and Method Choice
Select integration rules by interpreting accumulation and recognizing reverse-derivative structure, including linearity, powers, substitution, and definite-integral bounds.
- 09supplement
Supplement: Common Antiderivatives
Build a reliable antiderivative reference from derivative checks, including powers, exponentials, logarithms, trigonometric functions, and constants of integration.
Course Synthesis
1 item0L1A
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Mathematics · Subject
Calculus II
7 units · 38 lessons · 0 articles
Unit 1 - Integration Foundations and Strategic Choice
5 items5L0A
- 01lesson
Substitution as the Reverse Chain Rule
Recognize composition in an integrand, make a legitimate change of variable, and verify the resulting antiderivative.
- 02lesson
Antiderivatives, Definite Integrals, and Net Change
Interpret a volumetric flow-rate graph, distinguish net change from total flow, and evaluate definite integrals with units.
- 03lesson
Definite Substitution and Changing Bounds
Keep a new variable and its interval consistent when evaluating a definite integral by substitution.
- 04lesson
Method Recognition and Integral Triage
Read an integrand's structure, select a justified first move, and test whether that move simplifies the problem.
- 05lesson
Numerical Integration and Error
Estimate accumulated change from measured values and explain what the estimate can and cannot claim.
Unit 3 - Accumulation in Geometry, Physics, and Models
6 items6L0A
- 01lesson
Area Between Curves and Changing Roles
Model a bounded region with vertical or horizontal slices, locate intersections, and split an integral when the bounding functions exchange roles.
- 02lesson
Volume by Cross-Sections and Washers
Turn a solid into thin cross-sections, identify outer and inner radii, and accumulate cross-sectional area into volume.
- 03lesson
Cylindrical Shells and Method Choice
Build volumes from thin cylindrical shells and choose deliberately between shells and washers.
- 04lesson
Arc Length and Surface Area
Approximate a curve by short straight segments, then use the limiting geometry to find arc length and surfaces of revolution.
- 05lesson
Work, Fluid Force, and Variable Density
Model a changing force or weight one thin piece at a time and accumulate the physical contributions with an integral.
- 06lesson
Moments, Centers, and Probability
Use weighted accumulation to find a center of mass, centroid, and continuous expected value.
Unit 2 - Integration Methods as Structural Rewriting
6 items6L0A
- 01lesson
Integration by Parts from the Product Rule
Derive integration by parts from the product rule and choose its parts because the new integral becomes simpler.
- 02lesson
Repeated and Cyclic Integration by Parts
Use repeated integration by parts when it reduces an algebraic factor, and solve a cyclic integral by returning the original integral to one side.
- 03lesson
Trigonometric Integrals
Use parity and Pythagorean identities to rewrite powers of sine and cosine into integrable forms.
- 04lesson
Trigonometric Substitution
Match a quadratic radical to a Pythagorean identity, use a reference triangle to return to the original variable, and check domain restrictions.
- 05lesson
Partial Fractions and Algebraic Decomposition
Decompose a proper rational function into simpler fractions and verify the algebra before integrating.
- 06lesson
Technique Synthesis and Verification
Select, combine, and verify integration techniques by following structural evidence rather than a memorized formula list.
Unit 4 - Differential Equations and Alternative Coordinates
5 items5L0A
- 01lesson
Separable Differential Equations
Recognize when a rate law can be separated, integrate with conditions, and interpret the resulting family of models.
- 02lesson
Qualitative Models and Numerical Approximation
Read a differential equation through slopes, equilibrium behavior, and Euler’s method before seeking an exact formula.
- 03lesson
Parametric Curves and Motion
Describe position with paired functions of a parameter, then read direction, velocity, and speed from the representation.
- 04lesson
Polar Coordinates and Graphing
Use distance and direction to plot polar points, interpret negative radius, and trace polar curves deliberately.
- 05lesson
Polar Area, Length, and Representation Choice
Accumulate polar sectors and arc elements while using curve tracing to select correct bounds and representations.
Unit 5 - Sequences and Series Foundations
6 items6L0A
- 01lesson
Sequences and Their Limits
Treat a sequence as an ordered process, use numerical and symbolic evidence responsibly, and express convergence precisely.
- 02lesson
Series, Partial Sums, and Geometric Series
Distinguish a list of terms from their accumulated sum, then use geometric structure to decide and compute convergence.
- 03lesson
Divergence and the Integral Test
Use the necessary term condition and the area under a related positive function to make defensible convergence decisions.
- 04lesson
Comparison Tests
Compare positive-term series to known benchmarks, choosing direct or limit comparison from the structure of the terms.
- 05lesson
Alternating and Absolute Convergence
Determine when alternating signs create a convergent series and distinguish conditional convergence from absolute convergence.
- 06lesson
Ratio and Root Tests
Use successive-term ratios and nth roots to classify series controlled by factorial or exponential structure.
Unit 6 - Power Series and Taylor Models
6 items6L0A
- 01lesson
Power Series and Intervals of Convergence
Treat a power series as a family of numerical series, use the ratio test for its radius, and test endpoints separately.
- 02lesson
Operating on Power Series
Add, multiply, differentiate, and integrate power series while tracking the interval where the new representation is valid.
- 03lesson
Taylor Polynomials from Derivative Matching
Construct a local polynomial approximation by matching a function’s value, slope, curvature, and higher derivative information at a center.
- 04lesson
Maclaurin Series for Key Functions
Recognize and derive core Maclaurin series, then use substitution and algebra to create reliable local representations.
- 05lesson
Remainders and Error Bounds
Turn a Taylor approximation into a defensible numerical claim by estimating the remainder under stated derivative bounds.
- 06lesson
Approximation and Modeling Synthesis
Select a representation, build an approximation, and report a result with assumptions, units, and an explicit error claim.
Unit 7 - Calculus II Synthesis and the Calculus III Bridge
4 items4L0A
- 01lesson
Integration Strategy Across Representations
Choose an integration method by reading structure, geometry, and representation before beginning symbolic manipulation.
- 02lesson
Modeling Accumulated Change
Build definite-integral models from rates, state variables, units, and sign conventions, then interpret the result responsibly.
- 03lesson
Convergence Strategy and Approximation Judgment
Select convergence tests from structure, distinguish proof from computation, and make approximation decisions with stated error evidence.
- 04lesson
Calculus II Synthesis and Calculus III Bridge
Connect the course’s representations, accumulation models, infinite processes, and approximation habits to the questions of multivariable calculus.
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