Ontario Grade 12 MCV4U Calculus and Vectors (Grade 12, University Preparation)
Chapters: 3
1. A. Rate of Change
1 Investigating Instantaneous Rate of Change at a Point · 2 Investigating the Concept of the Derivative Function · 3 Investigating the Properties of Derivatives
- Limits (Class 11): What a Function Gets Close To – A limit tells us which number f(x) gets close to when x gets close to a point. We walk towards the point from the left and from the right. If both sides reach the same number, that number is the limit. You will learn limits of polynomial, rational, trigonometric, exponential and log functions, with the standard results you must know.
- Derivatives (Class 11): Slope, Rate of Change and Rules – A derivative tells how fast something changes at one moment. On a graph it is the slope of the tangent line. We find it with a limit: take a tiny step h, find the average change, and let h go to 0. Then we learn the quick rules for xⁿ, sin x, cos x, sums, differences, products and quotients.
2. B. Derivatives and Their Applications
1 Connecting Graphs and Equations of Functions and Their Derivatives · 2 Solving Problems Using Mathematical Models and Derivatives
- Application of Derivatives – The derivative measures how fast one quantity changes compared with another. Its sign tells whether a function goes up (f′ > 0, increasing) or down (f′ < 0, decreasing). Where f′ = 0 the tangent is flat: these critical points may be a local maximum or minimum, checked by the first derivative test (sign change) or the second derivative test (sign of f″). This lets us solve real problems like the biggest box or the cheapest tank.
3. C. Geometry and Algebra of Vectors
1 Representing Vectors Geometrically and Algebraically · 2 Operating With Vectors · 3 Describing Lines and Planes Using Linear Equations · 4 Describing Lines and Planes Using Scalar, Vector, and Parametric Equations
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- Lines and Planes in Space – Two points fix a line; three points not on one line fix a plane. Two lines in space meet, are parallel or are skew. A line lies in a plane, is parallel to it or cuts it at one point; two planes are parallel or meet in a line. A line is perpendicular to a plane if it is perpendicular to two crossing lines of the plane. With normal vector n = (a, b, c) the plane is ax + by + cz = d, a line is r = a + t·u, and the distance from (x₀, y₀, z₀) to the plane is |ax₀ + by₀ + cz₀ − d| ÷ √(a² + b² + c²).
- Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.