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Calculus and Vectors

Grade 12 University Mathematics, following the Ontario curriculum.

  • 6 units
  • 29 lessons
  • 282 practice questions
  • 215 flashcards
  • 4 simulators
Start Unit 1Exam review →
Try oneMCV4U · Average and Instantaneous Rates of Change
Find the average rate of change of from to .

Units

Unit 1 · 4 lessons · ~10% of course

Rates of Change & Limits

How fast is something changing right now? Secants give average rates over an interval, limits shrink that interval to a single point, and the result is the derivative, the idea the rest of calculus is built on.

  1. 1Average and Instantaneous Rates of Change
  2. 2Limits: Graphs, Tables, Sequences and Continuity
  3. 3The Derivative from First Principles
  4. 4The Derivative Function
Unit 2 · 6 lessons · ~20% of course

Derivative Rules

Stop taking limits by hand: the power, sum, product, chain and quotient rules, then the derivatives of sine, cosine, eˣ and aˣ, so any function in the course can be differentiated in a few lines.

  1. 1Power, Constant, Sum and Difference Rules
  2. 2The Product Rule
  3. 3The Chain Rule
  4. 4Derivatives of Rational Functions
  5. 5Derivatives of Sine and Cosine
  6. 6Derivatives of Exponential Functions
Unit 3 · 4 lessons · ~15% of course

Curve Sketching

Use the first and second derivatives to find exactly where a graph rises, falls, turns and bends, then put it all together to sketch polynomial and simple rational functions, and to read a function's story from the graph of its derivative.

  1. 1Increasing and Decreasing Functions and the First Derivative Test
  2. 2Concavity, Points of Inflection and the Second Derivative Test
  3. 3Sketching Polynomial and Simple Rational Functions
  4. 4Graphs of f, f′ and f″
Unit 4 · 4 lessons · ~15% of course

Applications of Derivatives

Put the derivative to work: velocity and acceleration of moving objects, instantaneous rates in population, decay, temperature, daylight, tides and prices, and optimization problems that find the biggest box, the cheapest can and the fare that earns the most.

  1. 1Motion: Displacement, Velocity and Acceleration
  2. 2Rates of Change in Context
  3. 3Optimization Problems
  4. 4Optimization in Business and Science
Unit 5 · 6 lessons · ~20% of course

Vectors

Quantities with a direction: describe them three ways, add and scale them, write them as components in 2-space and 3-space, steer planes through the wind, and multiply them with the dot and cross products.

  1. 1Vectors, Scalars and Direction
  2. 2Adding, Subtracting and Scaling Vectors
  3. 3Cartesian Vectors in 2-Space and 3-Space
  4. 4Applications: Forces, Wind and Current
  5. 5The Dot Product
  6. 6The Cross Product
Unit 6 · 5 lessons · ~20% of course

Lines & Planes

Describe lines and planes with scalar, vector and parametric equations, sort out how lines and planes meet in 3-space, solve systems of three equations as three planes, and measure the shortest distances between them.

  1. 1Lines in 2-Space
  2. 2Lines in 3-Space
  3. 3Planes: Normals and Equations
  4. 4Intersections of Lines and Planes
  5. 5Distances in 3-Space

Simulators

Simulators for every course →
MCV4U
Derivative Grapher

Slide a point along a function to watch its tangent slope, trace the graph of f′ point by point, and see where f′ and f″ reveal maxima, minima and inflection points.

Used in 7 lessons
MCV4U
Optimization Lab

Build a box, fence a field, design a can or price a ticket, then use the derivative to find the size or price that gives the best result.

Used in 2 lessons
MCV4U
Vector Lab

Drag two vectors to see their components, magnitude, angle and bearings, add and subtract them tip-to-tail or by parallelogram, and watch the dot product and projections change.

Used in 2 lessons
MCV4U
Plane Explorer

Rotate a 3-D view of up to three planes and a line from their equations to see where they meet (a point, a line, or nowhere) and solve the system step by step.

Used in 2 lessons