Calculus and Vectors (Grade 12)
39 of 39 topics written so far. Greyed-out topics are coming soon.
Unit 1: Rates of Change and Limits
- 1Instantaneous Rate of Change
Estimating the rate of change at a single moment as the slope of a tangent, using secants over shrinking intervals and centred intervals.
Draft - 2Introduction to Limits
The idea of a limit — what a function approaches near a point, limit notation, one-sided limits, and when a limit does not exist.
Draft - 3Estimating Limits from Graphs and Tables
Reading limits and one-sided limits from a graph, estimating limits from tables of values, and checking that graphs, tables, and equations agree.
Draft - 4Algebraic Techniques for Limits
Evaluating 0/0 limits by factoring, rationalizing with conjugates, and combining fractions, and choosing the right procedure for a limit.
Draft - 5The Definition of the Derivative
The derivative as a limit, derivative notation, tangent lines, and estimating derivatives from tables and graphs.
Draft - 6Differentiability and Continuity
Why differentiable functions are continuous, where derivatives fail to exist (corners, cusps, vertical tangents, discontinuities), and checking piecewise functions.
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Unit 2: Derivative Rules
- 1The Power Rule and Basic Derivative Rules
Differentiating powers of x (including negative and fractional exponents), constants, sums, differences, and constant multiples.
Draft - 2The Product Rule
Differentiating a product of two (or three) functions, including from tables of values and in context.
Draft - 3The Chain Rule
Differentiating composite functions with f'(g(x)) · g'(x) and dy/du · du/dx — powers, trig, exponential, and log compositions, tables of values, and choosing which rules to combine.
Draft - 4The Quotient Rule and Trig Derivatives
Differentiating quotients, and using the quotient rule to find the derivatives of tan x, cot x, sec x, and csc x.
Draft - 5Derivatives of Sine, Cosine, eˣ, and ln x
The derivatives of sin x, cos x, e^x, and ln x, why radians matter, and using them with the basic derivative rules.
Draft - 6Derivatives of Exponential Functions
Why the derivative of a^x is a constant times a^x, how that leads to the number e, the rule d/dx a^x = a^x ln a, the chain rule with e^(kx) and A·b^(kt), and rates of growth and decay.
Draft - 7Higher-Order Derivatives
Second, third, and higher derivatives — notation like f''(x), d²y/dx², and y'', what the second derivative means, and second derivatives of implicit relations.
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Unit 3: Curve Sketching
- 1Critical Points and Extrema
Absolute and relative maximums and minimums, the Extreme Value Theorem, and finding critical points where f'(x) = 0 or f'(x) does not exist.
Draft - 2First Derivative Test
Finding where a function increases and decreases from the sign of f', building sign charts, and using the first derivative test to classify relative extrema.
Draft - 3Concavity and the Second Derivative Test
Concave up and concave down from the sign of f'', points of inflection, and the second derivative test for relative extrema — including when it's inconclusive.
Draft - 4Connecting f, f′, and f″
Reading the features of f from graphs of f' and f'', and sketching f and f' from each other — increasing, extrema, concavity, and inflection points.
Draft - 5Curve Sketching
A complete step-by-step method for sketching polynomial and simple rational functions from their equations — domain, intercepts, symmetry, asymptotes, f' and f'' sign charts, and a summary table.
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Unit 4: Applications of Derivatives
- 1Straight-Line Motion with Derivatives
Position, velocity, and acceleration of a particle on a line — speed, direction, changing direction, and speeding up versus slowing down.
Draft - 2Derivatives in Context
Interpreting a derivative as a rate of change with units, and writing AP-style sentences about what f'(a) means in a real situation.
Draft - 3Optimization
Setting up and solving optimization problems with derivatives — variables, constraint, objective function, domain — and justifying the answer for fencing, boxes, cans, cost, and distance.
Draft - 4Optimization with Rational and Exponential Models
Solving optimization problems with rational and exponential models — average cost, revenue from exponential demand, drug concentration, and maximizing an average rate — with domain checks and justification.
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Unit 5: Vectors
- 1Introduction to Vectors
Scalars vs vectors, magnitude and direction, vector notation, equal and opposite vectors, and describing directions with bearings and angles.
Draft - 2Adding and Subtracting Vectors
Adding vectors tip to tail and with the parallelogram law, subtracting by adding the opposite, the zero vector, properties of addition, and finding a resultant with the cosine and sine laws.
Draft - 3Scalar Multiplication of Vectors
Multiplying a vector by a number, collinear vectors, unit vectors, the distributive properties, linear combinations, and simple geometric proofs with vectors.
Draft - 4Applications of Vectors
Resultant and equilibrant forces, and navigation problems with wind and current (heading vs track, air speed vs ground speed), solved with vector diagrams and the sine and cosine laws.
Draft - 5Cartesian Vectors
Writing 2-D vectors in component form [x, y], position vectors, converting between components and magnitude-direction, the vector between two points, unit vectors i and j, and operations in component form.
Draft - 6Vectors in Three Dimensions
The right-handed 3-D coordinate system, plotting points, vectors [x, y, z] and the unit vectors i, j, k, magnitude, distance between points, operations in component form, and collinearity.
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Unit 6: Dot and Cross Products
- 1The Dot Product
Multiplying two vectors to get a number — the geometric and component formulas, the angle between vectors, perpendicular vectors, properties, and work.
Draft - 2Vector Projections
The scalar and vector projection of one vector onto another, splitting a vector into parallel and perpendicular parts, direction cosines, and applications to forces and work.
Draft - 3The Cross Product
Multiplying two vectors in 3-space to get a perpendicular vector — the component formula, the determinant shortcut, the right-hand rule, the magnitude formula, and properties.
Draft - 4Applications of the Cross Product
Using the cross product to find areas of parallelograms and triangles, volumes with the scalar triple product, torque on a wrench, and vectors normal to two given vectors.
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Unit 7: Lines and Planes
- 1Linear Equations in 2-Space and 3-Space
What the solutions of linear equations look like — lines in 2-space, planes in 3-space — and how systems of two equations meet in a point, a line, or not at all.
Draft - 2Equations of Lines in 2-Space
Vector, parametric, and scalar equations of a line in the plane, normal and direction vectors, converting between forms, and parallel and perpendicular lines.
Draft - 3Equations of Lines in 3-Space
Vector, parametric, and symmetric equations of a line in 3-space, why a line has no single scalar equation, and writing a line as the intersection of two planes.
Draft - 4Equations of Planes
Normal vectors, scalar, vector, and parametric equations of a plane, converting between them with the cross product, the plane through three points, and parallel and perpendicular planes.
Draft - 5Intersections of Lines and Planes
How two lines in 3-space can meet (intersecting, parallel, coincident, or skew), how a line meets a plane, and how to find the point of intersection.
Draft - 6Intersections of Planes
How two or three planes can meet — in a point, a line, a plane, or not at all — solving systems of three equations by elimination, and using normals to predict the result.
Draft - 7Distances to Lines and Planes
Shortest distances from a point to a line or a plane, between parallel planes, and between skew lines, derived from projections and cross products.
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