Calculus 12

Unit 1: Calculus 12 — Rates of Change and the Derivative

Average and instantaneous rates of change, the secant-to-tangent limit, the formal definition of the derivative, and implicit differentiation.

• Average vs. instantaneous rate of change
• From secant slopes to the tangent slope

• The derivative as a limit
• Where a function is not differentiable

• Implicit differentiation

Unit 2: Calculus 12 — Differentiation Rules and Applications

Limits and continuity of combined functions, the differentiation rules, derivatives of trigonometric, exponential and logarithmic functions, and applications: rates of change, related rates, differentials, optimization, and definite integrals.

• Calculating limits and their properties
• Removing removable discontinuities
• Continuity of combinations and composites

• Power, sum, and difference rules
• Product and quotient rules
• The chain rule
• Derivatives of trigonometric functions
• Derivatives of exponential and logarithmic functions

• Calculating and interpreting rate of change
• Using derivatives to solve rate-of-change problems
• Estimating change with differentials
• Related rates
• Critical points and absolute extrema
• Increasing and decreasing intervals
• Optimization: maximum and minimum values

• Rules for definite integrals
• The Fundamental Theorem of Calculus
• Integration by substitution
• Integration by parts
• Net change from an integrated rate

Unit 3: Calculus 12 — Curve Analysis, Tangents and Antiderivatives

Continuity of functions, the tangent-from-secant development, tangent and normal lines, the graphical connection between f and f', the derivative tests, concavity and inflection, initial value problems, and antiderivatives via the Fundamental Theorem.

• Intervals of continuity

• Developing the tangent from the secant
• Equations of tangent and normal lines

• The connection between the graphs of f and f'
• First and Second Derivative Tests for local extrema
• Concavity and points of inflection

• Initial value problems
• The derivative–definite integral relationship
• Constructing antiderivatives with the FTC
• Antiderivatives of polynomials
• Antiderivatives of exponential and trigonometric functions

Unit 4: Calculus 12 — Integration and Area

Riemann sums, the meaning of area under a curve, the definite integral as area, numerical integration, areas between curves, and volumes of revolution.

• Riemann sums for area under a curve
• The meaning of area under the curve

• Expressing area as a definite integral
• Computing area by numerical integration

• Areas of regions in a plane
• Volumes by slices or shells