Calculus and Vectors, Grade 12, University Preparation
Curriculum Policy
Mathematics, The Ontario Curriculum, Grades 11 and 12, 2007 (Revised)

Course Description:
This course enhances students’ foundational knowledge of functions while deepening their understanding of rates of change. Students will engage with geometric and algebraic representations of vectors, as well as lines and planes in three-dimensional space. The curriculum broadens their insights into rates of change, introducing them to the derivatives of various types of functions, including polynomial, sinusoidal, exponential, rational, and radical functions. Through these concepts, students will apply their learning to model and analyze real-world relationships, honing the mathematical processes required for success in advanced mathematics. This course is tailored for students aiming for careers in fields such as science, engineering, economics, and specific business sectors, serving as an important foundation for those planning to take university-level courses in calculus, linear algebra, or physics.
Chapters and Descriptions:

• Introduction to Vectors
  This introductory unit explores four primary topics: distinguishing between vectors and scalars, understanding vector properties, performing vector operations, and examining the properties of plane figures. Students will learn to identify scalar and vector quantities, represent vectors as directed line segments, and carry out operations such as addition, subtraction, and scalar multiplication on geometric vectors, both with and without dynamic geometry software. The unit culminates with students proving certain properties of plane figures using vector methods and solving problems related to force and velocity.
• Linear Independence and Coplanarity
  In this unit, students will delve into Cartesian vectors represented as ordered pairs and triples in two-dimensional and three-dimensional spaces, respectively. The focus will be on addition, subtraction, and scalar multiplication of these vectors. Students will explore linear dependence, independence, collinearity, and coplanarity concepts to enhance their understanding of vector relationships.
• Vector Applications
  This unit introduces students to practical applications involving work and torque, providing context for understanding the dot and cross products of Cartesian vectors. Students will express vector and scalar projections in terms of the dot product, while investigating and proving the properties of these vector products. The applications will prepare students to predict characteristics of solutions in systems of lines and planes.
• Lines and Planes in 3D Space
  Students will begin this unit by deriving vector, parametric, and symmetric equations for lines in both R2R^2R2 and R3R^3R3. The exploration continues with determining the equations for planes in three-dimensional space. Students will analyze the intersections of lines and planes in R3R^3R3 and learn to solve systems of linear equations involving two or three planes. The unit emphasizes the geometric interpretation of solutions to linear systems, alongside matrix operations, including addition, subtraction, and multiplication, and introduces row reduction techniques for solving systems of linear equations.
• Calculus Fundamentals
  This unit focuses on essential mathematical operations required for calculus. Students will develop an understanding of rates of change and the concept of limits. The unit will cover the fundamental idea that a limit approaches a value without necessarily reaching it, as well as techniques for determining limits through substitution, factoring, rationalization, and change of variables. Students will explore the relationship between secant and tangent lines and derive a tangent slope function, culminating in graphing derivative functions.
• Understanding Derivatives
  In this unit, students will learn about derivatives as shortcuts for determining tangent line slopes. The unit begins with the exploration of key rules for derivatives, including the power, product, quotient, and chain rules, along with derivatives of composite functions. Students will also apply these rules to find higher-order derivatives and relate position, velocity, and acceleration functions through differentiation.
• Curve Analysis and Sketching
  Building on previous graphing techniques, this unit emphasizes the importance of calculus in accurately sketching curves. Students will review key features that contribute to a well-formed graph before integrating these elements into comprehensive curve sketches.
• Real-World Applications of Derivatives
  This unit focuses on various problem types involving derivatives, categorized into Pythagorean Theorem problems, volume problems, trough problems, shadow problems, and general rate problems. Students will investigate each problem type, gaining practical experience in applying derivative concepts.
• Exponential and Logarithmic Differentiation
  Students will explore derivatives of exponential and logarithmic functions, beginning with Euler’s number eee and extending to other bases. The unit will also cover techniques for sketching curves involving these functions and introduce logarithmic differentiation for functions whose derivatives are not easily determined using previously established rules.
• Differentiation of Trigonometric Functions
  The final unit begins with a review of trigonometric concepts and progresses to special angles and the CAST rule for identifying the positivity and negativity of basic trigonometric ratios in different quadrants. Students will solve trigonometric equations using the CAST rule and investigate fundamental trigonometric limits, concluding with an assignment and quiz to assess their understanding.