The concept of a limit is foundational to calculus.
Differential calculus develops the concept of instantaneous rate of change.
Integral calculus develops the concept of determining a product involving a continuously changing quantity over an interval.
Derivatives and integrals are inversely related.
Limits & Continuity
piecewise defined functions
finding limits graphically
finding limits algebraically
continuity
limits involving infinity
The Derivative
tangent lines
the derivative
differentiation techniques: power rule, product and quotient rules, chain rule
implicit differentiation
higher order differentiation
Applications of the Derivative
curve analysis
optimization
related rates
growth and decay
Transcendental Functions
derivatives of trig functions
derivatives of exponential functions
derivatives of logarithmic functions
Integration
antiderivatives
definite and indefinite integral
integration techniques
Applications of Integration
finding area under a curve
finding area between curves
volume of solids of revolution
Reasoning and Analyzing
estimate reasonably
use reason and logic to analyze and apply mathematical ideas
use tools or technology to analyze relationships, test conjectures, and check solutions
Understanding and Solving
develop a conceptual understanding of ideas
visualize to explore and illustrate concepts and relationships
apply flexible strategies to solve problems
Communicating and Representing
communicate mathematical thinking in many ways
use mathematical language to contribute to mathematical discussions
represent ideas in various ways (written, symbolic, pictorial)
explain and justify ideas
Connecting and Reflecting
reflect on mathematical thinking
connect mathematical concepts to each other