Ontario Grade 12 University Preparation math (MCV4U)

Limits, derivatives, rules, and rates of change.

• A1.1 · Interpret rate-of-change applications and representations (written lesson not published)
• A1.2 · Connect secant and tangent slopes with rates (written lesson not published)
• A1.3 · Approximate instantaneous rate with approaching secants (written lesson not published)
• A1.4 · Interpret limits graphically and numerically (written lesson not published)
• A1.5 · Use the difference quotient for rates of change (written lesson not published)
• A1.6 · Compare simplified and unsimplified first-principles derivatives (written lesson not published)
• A2.1 · Use derivative signs to describe intervals (written lesson not published)
• A2.2 · Generate derivative values and graphs (written lesson not published)
• A2.3 · Differentiate polynomial functions from first principles (written lesson not published)
• A2.4 · Relate sine and cosine functions to their derivatives (written lesson not published)
• A2.5 · Investigate derivatives of exponential functions (written lesson not published)
• A2.6 · Identify the special derivative property of e to the x (written lesson not published)
• A2.7 · Connect natural logarithms with the inverse of e to the x (written lesson not published)
• A2.8 · Differentiate exponential functions with any base (written lesson not published)
• A3.1 · Verify the power rule for natural-number exponents (written lesson not published)
• A3.2 · Apply constant, multiple, sum, and difference rules (written lesson not published)
• A3.3 · Differentiate polynomials and solve rate problems (written lesson not published)
• A3.4 · Extend the power rule to rational exponents (written lesson not published)
• A3.5 · Apply product and chain rules to composite functions (written lesson not published)

Curve sketching, motion, modelling, and optimization.

• B1.1 · Sketch a derivative graph from a function graph (written lesson not published)
• B1.2 · Interpret the second derivative as slope change (written lesson not published)
• B1.3 · Connect features of a function and its first two derivatives (written lesson not published)
• B1.4 · Infer possible functions from derivative information (written lesson not published)
• B1.5 · Sketch polynomial curves using derivatives (written lesson not published)
• B2.1 · Model displacement, velocity, and acceleration (written lesson not published)
• B2.2 · Solve applications using derivatives (written lesson not published)
• B2.3 · Solve instantaneous-rate problems (written lesson not published)
• B2.4 · Solve optimization problems (written lesson not published)
• B2.5 · Apply and interpret calculus models (written lesson not published)

Vector operations and equations of lines and planes.

• C1.1 · Interpret vector magnitude, direction, and applications (written lesson not published)
• C1.2 · Represent two-dimensional vectors in Cartesian and polar form (written lesson not published)
• C1.3 · Convert two-dimensional vector representations (written lesson not published)
• C1.4 · Represent points and vectors in three dimensions (written lesson not published)
• C2.1 · Add, subtract, and multiply vectors by scalars (written lesson not published)
• C2.2 · Verify properties of vector operations (written lesson not published)
• C2.3 · Solve applications with vector operations (written lesson not published)
• C2.4 · Calculate and interpret the dot product (written lesson not published)
• C2.5 · Apply properties of the dot product (written lesson not published)
• C2.6 · Calculate and interpret the cross product (written lesson not published)
• C2.7 · Apply properties of the cross product (written lesson not published)
• C2.8 · Solve applications with dot and cross products (written lesson not published)
• C3.1 · Interpret intersections of lines and systems in two dimensions (written lesson not published)
• C3.2 · Interpret intersections of planes and systems in three dimensions (written lesson not published)
• C3.3 · Analyze geometric configurations of lines and planes (written lesson not published)
• C4.1 · Write scalar, vector, and parametric equations of lines in two dimensions (written lesson not published)
• C4.2 · Write vector and parametric equations of lines in three dimensions (written lesson not published)
• C4.3 · Use normal vectors to describe planes (written lesson not published)
• C4.4 · Solve systems representing three planes (written lesson not published)
• C4.5 · Derive scalar, vector, and parametric equations of planes (written lesson not published)
• C4.6 · Convert among equations of lines and planes (written lesson not published)
• C4.7 · Solve line-plane intersection and distance problems (written lesson not published)

Differentiate f(x) = 3x⁴ − 2x² + 5.

Worked method: Apply the power rule term by term: f′(x) = 12x³ − 4x.

For a = (1, 2) and b = (3, 0), find a · b.

Worked method: Multiply corresponding components and add: a · b = 1(3) + 2(0) = 3.

Differentiate y = (2x − 1)³.

Find the magnitude of v = (2, −3, 6).