Calculus and Vectors, Grade 12 (University)

Unit 1: Rate of Change

Investigating instantaneous rate of change through secants and limits, the concept of the derivative function, and verifying the rules for determining derivatives.

• Real-world examples of rates of change
• Connecting average rate of change to secants and instantaneous rate to tangents
• Approximating instantaneous rate of change using secants approaching the tangent
• Recognizing limits graphically and numerically
• Connecting average and instantaneous rate of change to the limit expression
• Comparing simplified and unsimplified instantaneous rate calculations
• Determining intervals where instantaneous rate of change is positive, negative, or zero

• Graphing the derivative from a table of instantaneous rates of change
• Determining polynomial derivatives using the limit definition
• Investigating the derivative graph of a sinusoidal function
• Investigating the derivative graph of an exponential function
• Discovering the number e as the base where f'(x) = f(x)
• The natural logarithm as the inverse of the exponential function e^x
• Verifying the derivative of a general exponential function

• Verifying the power rule for natural-number exponents
• Verifying the constant, constant multiple, sum, and difference rules
• Determining polynomial derivatives algebraically and using them for rate-of-change problems
• Verifying the power rule for rational exponents
• Verifying the chain rule
• Verifying the product rule
• Solving derivative problems using the product and chain rules

Unit 2: Derivatives and Their Applications

Connecting graphs and equations of functions and their derivatives, curve sketching using first and second derivatives, and solving optimization and rate-of-change problems using derivatives.

• Sketching a derivative graph and recognizing points of inflection
• The second derivative as the rate of change of the rate of change
• Determining the second derivative algebraically
• Connecting the graphs of a function and its first and second derivatives
• Sketching multiple possible functions from derivative information

• Sketching a polynomial graph using first/second derivative strategies
• Connecting displacement, velocity, and acceleration to the derivative
• Connecting derivatives to real-world applications
• Solving instantaneous rate-of-change problems using the derivative
• Solving optimization problems
• Solving real-world problems using a mathematical model and the derivative

Unit 3: Geometry and Algebra of Vectors

Representing and operating with vectors in two-space and three-space, and representing and solving problems involving lines and planes in three-dimensional space.

• A vector as a quantity with magnitude and direction
• Representing a vector in two-space geometrically and algebraically
• Converting between Cartesian form and directed line segment form for two-space vectors
• Representing points and vectors in three-space and finding distance and magnitude

• Adding, subtracting, and scalar-multiplying vectors
• Properties of vector addition, subtraction, and scalar multiplication
• Solving real-world problems with vector addition, subtraction, and scalar multiplication
• The dot product and its applications
• Properties of the dot product
• The cross product and its applications
• Properties of the cross product
• Solving real-world problems involving the dot and cross product

• Solution points of linear equations in two-space and their intersection
• Solution points of linear equations in three-space as planes and their line of intersection
• Geometric configurations of up to three lines and/or planes in three-space

• Scalar, vector, and parametric equations of a line in two-space
• Representing a line in three-space
• The normal to a plane and geometric properties of a plane
• The scalar equation of a plane and finding the intersection of three planes
• Determining the equations of a plane from its properties
• Converting between forms of a plane's equation
• Solving problems involving distances and intersections of lines and planes