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Calculus
1
A foundational yet rigorous introduction to differential and integral calculus. Master limits, derivatives, and integrals in a way that builds real intuition — not just formulas.
What You'll Learn
A full semester of Calculus 1 — completed at your own pace, with no deadlines or fixed class times.
What You'll Be Able to Do
- Evaluate functions using function notation, determine domains and ranges, and construct mathematical models using appropriate families of functions
- Explain and evaluate limits graphically, numerically, and analytically — including one-sided limits, limits at infinity, and the epsilon–delta definition — and determine continuity and classify discontinuities
- Interpret the derivative as a rate of change and slope of a tangent line, compute derivatives using the limit definition, and relate differentiability to continuity
- Apply differentiation rules — power, product, quotient, and chain — to algebraic, exponential, logarithmic, trigonometric, and inverse trigonometric functions, including implicit and logarithmic differentiation
- Use derivatives to locate and classify extrema, analyze concavity, sketch curves, apply the Mean Value Theorem, and solve optimization and related-rates problems
- Model exponential growth and decay and apply L'Hôpital's Rule to evaluate indeterminate forms
- Compute antiderivatives, approximate area with Riemann sums, evaluate definite integrals, apply the Fundamental Theorem of Calculus, and use substitution and integration to solve accumulation, net change, motion, and area-between-curves problems
Major Course Topics
Course chapters and lessons
Calculus 1 has 8 chapters across 49 instructional hours, as listed in the course syllabus. Expand any chapter to see the lessons inside it.
- Chapters
- 8
- Lessons
- 44
- Instructional hours
- 49 hrs
- Semester hours
- 4
Chapter 1
Functions and Mathematical Modeling
- 1.1 Function Basics Function notation and evaluation · Domain and range
- 1.2 Graphs and Behavior of Functions Interpreting graphs · Intercepts and symmetry · Transformations and key features
- 1.3 Function Families Polynomial and rational functions · Exponential and logarithmic functions · Trigonometric and inverse trigonometric functions
- 1.4 Constructing New Functions Sums, products, and quotients of functions · Compositions · Piecewise-defined functions
- 1.5 Modeling Real-World Scenarios Modeling using linear, exponential, and sinusoidal functions · Building models using transformations, compositions, inverses, and piecewise definitions · Interpreting models in context
Chapter 2
Limits and Continuity
- 2.1 Tangent and Velocity Problems (Motivation for Limits) Tangent line problem · Instantaneous velocity idea
- 2.2 Limits of a Function (Conceptual + Numerical + Graphical) Concept of a limit · Estimating limits from graphs and tables · One-sided limits
- 2.3 Computing Limits Algebraically Limit laws · Direct substitution · Algebraic simplification strategies · Squeeze (Sandwich) Theorem
- 2.4 Precise Definition of a Limit Epsilon-delta definition of a limit · Verifying limits using the epsilon-delta definition · Intuition behind the formal limit definition
- 2.5 Continuity Continuity at a point · Continuity on an interval/domain · Types of discontinuities (removable, jump, infinite) · Intermediate Value Theorem
- 2.6 Limits at Infinity and Asymptotes Limits as x → ∞ and x → −∞ · Horizontal and vertical asymptotes · End behavior of functions
Chapter 3
Derivatives and Differentiation
- 3.1 Rates of Change and the Derivative Concept Average rate of change vs instantaneous rate of change · Derivative as the slope of the tangent line · Derivative as a rate (motion interpretation)
- 3.2 The Derivative from First Principles Limit definition of the derivative · Derivatives of simple functions using the definition
- 3.3 The Derivative as a Function Interpreting derivative graphs · Building f′(x) from f(x) conceptually
- 3.4 Higher-Order Derivatives Second and higher derivatives · Notation and interpretation (acceleration, concavity)
- 3.5 Differentiability and Continuity Differentiability at a point and on an interval · Relationship: differentiability implies continuity · Common non-differentiable behaviors (corners, cusps, vertical tangents, discontinuities)
Chapter 4
Differentiation Rules and Techniques
- 4.1 Basic Derivative Rules Constant rule · Power rule · Constant multiple and sum/difference rules
- 4.2 Derivatives of Standard Functions Polynomial and rational functions · Exponential functions · Logarithmic functions · Trigonometric functions · Inverse trigonometric functions
- 4.3 Product and Quotient Rules Product rule · Quotient rule
- 4.4 Chain Rule Derivatives of composite functions · Nested compositions
- 4.5 Implicit Differentiation Implicitly defined functions · Differentiation with respect to x
- 4.6 Logarithmic Differentiation Differentiating products, quotients, and powers using logarithms · Functions of the form y = f(x)g(x)
Chapter 5
Applications of Derivatives
- 5.1 Maximum and Minimum Values Critical points · Local vs absolute extrema · The Extreme Value Theorem
- 5.2 Derivatives and Graph Behavior Increasing and decreasing tests · First derivative test · Concavity and inflection points · Second derivative test
- 5.3 Mean Value Theorem Rolle's Theorem · Mean Value Theorem · Geometric and physical interpretation of the theorem
- 5.4 Linear Approximation and Differentials Linearization of functions · Local linear approximation · Differentials and estimation
- 5.5 Newton's Method Iterative approximation of roots · Convergence and limitations
- 5.6 Curve Sketching Strategy for sketching graphs using derivatives · Combining tests into a full graph analysis
- 5.7 Optimization Problems Translating word problems into functions · Finding and justifying optimal solutions
- 5.8 Related Rates Setting up related rate relationships · Solving with implicit differentiation
- 5.9 Exponential Growth and Decay Differential equation dy/dt = ky · Modeling population growth and radioactive decay · Doubling time and half-life
- 5.10 L'Hôpital's Rule Indeterminate forms (0/0, ∞/∞) · Applying L'Hôpital's Rule · Repeated application and limits involving exponentials
Chapter 6
Integration and the Fundamental Theorem of Calculus
- 6.1 Antiderivatives Antiderivative concept · Basic antidifferentiation rules
- 6.2 Areas and Accumulation Area under a curve concept · Signed area interpretation
- 6.3 Riemann Sums Left, right, midpoint sums (conceptual + computational) · Approximating area/accumulation · Sigma notation and summation
- 6.4 The Definite Integral Definite integral as a limit of Riemann sums · Notation and interpretation · Properties of the definite
- 6.5 The Fundamental Theorem of Calculus FTC Part 1: Derivative of an integral · FTC Part 2: Evaluating definite integrals using antiderivatives
- 6.6 Average Value and Net Change Average value of a function on an interval · Net change interpretation of integrals
Chapter 7
Techniques of Integration
- 7.1 Indefinite Integrals Indefinite integral notation · Constant of integration
- 7.2 Substitution Rule (Intro) u-substitution · Recognizing substitution structure · Change of variables · Substitution in definite integrals
Chapter 8
Applications of Integration
- 8.1 Accumulation Functions Accumulation as total change over an interval · Integrals with variable upper limits
- 8.2 Net Change Theorem Net change as the integral of a rate function · Interpreting positive and negative accumulation
- 8.3 Distance and Displacement Relationship between position, velocity, and acceleration · Total distance traveled vs net displacement
- 8.4 Area of a Region Between Two Curves Using integration to find the area between curves · Integrating with respect to x and y
These chapters come straight from the official course syllabus, which also sets out the graded assignment and the proctored final exam.
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