Calculus and Vectors – MCV4U

About Course

Comprehensive review course for Ontario MCV4U (Grade 12 University Calculus and Vectors). Master the complete curriculum through structured lessons, worked examples, and self-assessment quizzes.

What you’ll learn:
– Rates of Change — limits, derivatives, and applications
– Derivatives of Polynomial, Rational, Exponential, Logarithmic, and Trigonometric Functions
– Curve Sketching — critical points, concavity, optimization
– Geometric Vectors — vector operations, dot and cross products
– Algebraic Vectors — vectors in R² and R³
– Lines and Planes — equations of lines and planes in 3D
– Intersections of Lines and Planes
– Applications to Physics, Engineering, and Computer Graphics

This course is designed for students preparing for:
– Final MCV4U exam
– University programs in Engineering, Computer Science, Physics, Mathematics, and Economics
– AP Calculus AB/BC preparation

Compute limits and apply the formal definition of the derivative
Master differentiation rules — power, product, quotient, and chain rules
Differentiate exponential, logarithmic, and trigonometric functions
Apply derivatives to optimization, related rates, and motion problems
Sketch curves using calculus — critical points, concavity, asymptotes
Perform vector operations in 2D and 3D — addition, scalar multiplication
Compute and apply the dot product, cross product, and scalar projections
Write equations of lines and planes in vector, parametric, and scalar forms
Solve intersection problems involving lines and planes in R³
Calculate distances from points to lines and planes
Prepare confidently for the MCV4U final exam and university-level calculus

Course Content

1: Introduction to Calculus
Limits, rates of change, and the birth of the derivative. Everything downstream rests on the difference quotient built here.

1.1 Radical Expressions: Rationalizing Denominators
1.2 The Slope of a Tangent
1.3 Rates of Change
1.4 The Limit of a Function
1.5 Properties of Limits
1.6 Continuity

2: Derivatives
From the limit definition to the differentiation rules: power, product, quotient and chain.

3: Derivatives and Their Applications
Second derivatives, motion, and optimization — where calculus starts paying for itself.

4: Curve Sketching
Reading a function's whole shape off its first and second derivatives.

5: Derivatives of Exponential and Trigonometric Functions
Two new function families, one new constant, and optimization applied to both.

6: Introduction to Vectors
The pivot from calculus to vectors — geometric first, then algebraic, then into three dimensions.

7: Applications of Vectors
Forces, velocities, and the two products — the tools every later chapter depends on.

8: Equations of Lines and Planes
Turning geometry into equations — vector, parametric, symmetric and Cartesian forms.

9: Relationships Between Points, Lines, and Planes
Every intersection case in space, solved systematically — and the distance formulas that finish the course.

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Course Content

1: Introduction to Calculus Limits, rates of change, and the birth of the derivative. Everything downstream rests on the difference quotient built here.

1.1 Radical Expressions: Rationalizing Denominators

1.2 The Slope of a Tangent

1.3 Rates of Change

1.4 The Limit of a Function

1.5 Properties of Limits

1.6 Continuity

2: Derivatives From the limit definition to the differentiation rules: power, product, quotient and chain.

2.1 The Derivative Function

2.2 The Derivatives of Polynomial Functions

2.3 The Product Rule

2.4 The Quotient Rule

2.5 The Derivatives of Composite Functions (Chain Rule)

3: Derivatives and Their Applications Second derivatives, motion, and optimization — where calculus starts paying for itself.

3.1 Higher-Order Derivatives, Velocity, and Acceleration

3.2 Maximum and Minimum on an Interval (Extreme Values)

3.3 Optimization Problems

3.4 Optimization Problems in Economics and Science

4: Curve Sketching Reading a function's whole shape off its first and second derivatives.

4.1 Increasing and Decreasing Functions

4.2 Critical Points, Local Maxima, and Local Minima

4.3 Vertical and Horizontal Asymptotes

4.4 Concavity and Points of Inflection

4.5 An Algorithm for Curve Sketching

5: Derivatives of Exponential and Trigonometric Functions Two new function families, one new constant, and optimization applied to both.

5.1 Derivatives of Exponential Functions, y = e^x

5.2 The Derivative of the General Exponential Function, y = b^x

5.3 Optimization Problems Involving Exponential Functions

5.4 The Derivatives of y = sin x and y = cos x

5.5 The Derivative of y = tan x

6: Introduction to Vectors The pivot from calculus to vectors — geometric first, then algebraic, then into three dimensions.

6.1 An Introduction to Vectors

6.2 Vector Addition

6.3 Multiplication of a Vector by a Scalar

6.4 Properties of Vectors

6.5 Vectors in R² and R³

6.6 Operations with Algebraic Vectors in R²

6.7 Operations with Vectors in R³

6.8 Linear Combinations and Spanning Sets

7: Applications of Vectors Forces, velocities, and the two products — the tools every later chapter depends on.

7.1 Vectors as Forces

7.2 Velocity

7.3 The Dot Product of Two Geometric Vectors

7.4 The Dot Product of Algebraic Vectors

7.5 Scalar and Vector Projections

7.6 The Cross Product of Two Vectors

7.7 Applications of the Dot Product and Cross Product

8: Equations of Lines and Planes Turning geometry into equations — vector, parametric, symmetric and Cartesian forms.

8.1 Vector and Parametric Equations of a Line in R²

8.2 Cartesian Equation of a Line

8.3 Vector, Parametric, and Symmetric Equations of a Line in R³

8.4 Vector and Parametric Equations of a Plane

8.5 The Cartesian Equation of a Plane

8.6 Sketching Planes in R³

9: Relationships Between Points, Lines, and Planes Every intersection case in space, solved systematically — and the distance formulas that finish the course.

9.1 The Intersection of a Line with a Plane and the Intersection of Two Lines

9.2 Systems of Equations

9.3 The Intersection of Two Planes

9.4 The Intersection of Three Planes

9.5 The Distance from a Point to a Line in R² and R³

9.6 The Distance from a Point to a Plane