Calculus I Foundations

Functions as Calculus Objects

Treat a function as an input-output assignment whose domain, range, units, and meaning must survive changes of representation.

Reading Graphs, Tables, and Function Models

Extract global and local behavior from graphs, tables, and formulas while respecting sampling limits, units, and model assumptions.

Requires: Functions as Calculus Objects

Transforming Functions with Purpose

Predict and explain vertical, horizontal, reflective, and scaling transformations from changes made inside and outside a parent function.

Requires: Reading Graphs, Tables, and Function Models

Parameters Across Function Families

Interpret parameters and invariant structure across linear, quadratic, exponential, logarithmic, and trigonometric function families.

Requires: Transforming Functions with Purpose

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.

Requires: Parameters Across Function Families

Difference Quotients as Movable Secants

Build, simplify, and interpret x-plus-h and x-minus-a difference quotients as families of secant slopes.

Requires: Average Rate of Change and Secant Slope

Algebraic Restructuring for Calculus

Use factoring, common denominators, and conjugates to reveal local structure while preserving domain restrictions and equivalence.

Requires: Difference Quotients as Movable Secants

Radians, Neighborhoods, and Calculus Readiness

Connect radians, absolute-value neighborhoods, exponent and logarithm laws, and domain discipline to the language calculus uses.

Requires: Algebraic Restructuring for Calculus

The Instantaneous-Rate Problem

Discover why ordinary average rates cannot directly measure change at one instant and why a controlled approach is mathematically necessary.

Requires: Radians, Neighborhoods, and Calculus Readiness

Secants Approaching a Local Rate

Investigate shrinking secant intervals from both directions and decide whether their slopes support one local-rate destination.

Requires: The Instantaneous-Rate Problem

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.

Requires: Secants Approaching a Local Rate

Evidence and Failure in Finite Limits

Evaluate finite-limit claims using graphs, tables, formulas, counterexamples, and calibrated distinctions among evidence, conjecture, and proof.

Requires: Reading Limit Notation and Nearby Behavior

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.

Requires: Evidence and Failure in Finite Limits

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.

Requires: One-Sided Limit Notation and Direction

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.

Requires: Directional Agreement and Two-Sided Existence

Computing Nontrivial Limits

Resolve indeterminate finite limits by selecting factoring, conjugates, or common denominators to expose valid nearby structure.

Requires: Limit Laws and Direct Substitution

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.

Requires: Computing Nontrivial Limits

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.

Requires: Making Approaches Precise: Epsilon and Delta

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.

Requires: Infinite Limits and Limits at Infinity

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.

Requires: Continuity and the Intermediate Value Theorem

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.

Requires: Derivative as Instantaneous Rate and Tangent Slope

Basic Derivative Rules

Derive and use the constant, sum, constant-multiple, and power rules while keeping each shortcut connected to the limit definition.

Requires: The Derivative as a Function

Product, Quotient, and Chain Rules

Differentiate changing products, quotients, and compositions by identifying their structure and tracking each linked rate of change.

Requires: Basic Derivative Rules

Implicit Differentiation and Inverse Functions

Find derivatives from relations that are not solved for y, then explain why inverse functions have reciprocal tangent slopes.

Requires: Product, Quotient, and Chain Rules

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.

Requires: Implicit Differentiation and Inverse Functions

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.

Requires: Linearization and Differentials

Extreme Values and Optimization

Use derivatives to identify and test extrema on a stated domain, then build and interpret a one-variable optimization model.

Requires: Higher Derivatives and Motion

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.

Requires: Extreme Values and Optimization

Graphing with First and Second Derivatives

Construct a defensible graph sketch from domain, anchors, derivative signs, extrema, and concavity rather than visual guesswork.

Requires: Mean Value Theorem and Consequences

Related Rates

Model simultaneous changes with a constraint equation, differentiate before substituting, and report the resulting rate with correct units.

Requires: Graphing with First and Second Derivatives

Newton's Method and Modeling Error

Use tangent-line iteration to approximate roots, check convergence responsibly, and distinguish numerical error from model limitations.

Requires: Related Rates

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.

Requires: Newton's Method and Modeling Error

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.

Requires: Antiderivatives and Simple Differential Equations

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.

Requires: Area, Sigma Notation, and Riemann Sums

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.

Requires: Accumulation Functions and Net Change

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.

Requires: The Fundamental Theorem of Calculus

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.

Requires: Substitution as Reverse Chain Rule

Area Between Curves

Model a bounded planar region with vertical or horizontal slices, determine the correct subtraction order, and split integrals when boundaries switch.

Requires: Numerical Integration and Average Value

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.

Requires: Area Between Curves

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.

Requires: Volumes by Slicing and Washers

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.

Requires: Work, Motion, and Total Change

Calculus I Course-Synthesis Narrative: Managing a Community Solar Array

A capstone story that asks learners to connect functions, limits, derivatives, and accumulation before opening a fully reasoned solution.

Requires: Calculus I Synthesis and Calculus II Readiness