Calculus and Vectors

This unit opens Grade 12 Calculus and Vectors (MCV4U) with the single idea that all of calculus is built on: the limit.

You will start with rates of change in words, tables, graphs and equations, and connect the average rate of change over an interval with the slope of a secant and the instantaneous rate of change at a point with the slope of a tangent. You will see why a smooth function (continuous, with no corners) has a tangent everywhere, and why there is no instantaneous rate at a corner. You will estimate instantaneous rates from average rates over shrinking intervals, from equations and from real data.

Then you will study limits carefully: limits of sequences (the Fibonacci ratios and the golden ratio, (1 + 1/n)ⁿ and the number e, infinite sums), limits of functions from graphs and tables, one-sided limits, limits involving infinity and asymptotes, the limit laws and continuity, and algebraic methods (factoring, expanding, common denominators, conjugates, changes of variable) for the indeterminate form 0/0.

Finally you will put the pieces together. The average rate of change over [a, a + h] is the difference quotient (f(a + h) − f(a))/h, and its limit as h → 0 is the instantaneous rate of change. You will compare calculating it with and without simplifying first, define the derivative at a point, f′(a), find it from first principles for polynomial, rational and radical functions, and decide where a function is differentiable.

Remember from MHF4U: Advanced Functions must be taken before or at the same time as this course. Its unit Rates of Change introduced secants, tangents, shrinking intervals and the difference quotient, and Combining Functions and Modelling built models of real situations; this unit makes those ideas exact. It also uses Polynomial Functions and Equations (factoring, the factor theorem) and Rational Functions and Equations (asymptotes), and from Grade 11 Functions (MCR3U) the units Sequences and Series and Functions and Equivalent Expressions.

Coming next: Derivatives and Differentiation Rules turns f′(a) into a derivative function and develops the power, product and chain rules; Derivatives of Exponential and Sinusoidal Functions uses the limits (sin x)/x → 1 and (1 + 1/n)ⁿ → e that you meet here. You need a scientific calculator (radian mode for trigonometric functions); a free graphing tool such as Desmos or GeoGebra helps but is not required. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007), Calculus and Vectors, Grade 12, University Preparation (MCV4U), strand A. Rate of Change, overall expectation A1 (demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit): specific expectations A1.1 to A1.6. Overall expectations A2 and A3 are taught in Derivatives and Differentiation Rules and Derivatives of Exponential and Sinusoidal Functions. The mathematical process expectations are woven through every lesson.

In this unit you will meet the central idea of calculus: the derivative. You will start with what the instantaneous rate of change tells you about a graph (where a function increases, decreases and turns), then build the derivative function f′ from tables of tangent slopes and see that the derivative of a polynomial is a polynomial of one lower degree. You will define the derivative exactly from first principles, f′(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h, and learn to read the graphs of f and f′ side by side.

Then the unit gets fast. You will learn the power rule (for natural-number, negative and rational exponents), the constant, constant multiple, sum and difference rules, and the product, quotient and chain rules. You will use them to differentiate polynomial, rational and radical functions, find tangent and normal lines and points with a given slope, compute second and higher derivatives, and solve rate-of-change problems about populations, costs, draining barrels, medicine levels and a boat passing a lighthouse.

Remember from MHF4U and earlier units: in Advanced Functions (MHF4U), the unit Rates of Change introduced secants, tangents and the difference quotient at a point; Polynomial Functions and Equations and Rational Functions and Equations gave you the factoring and graphing you will use here; and Combining Functions and Modelling introduced composition, which the chain rule differentiates. The unit Limits and Rates of Change in this course made limits precise. Rational exponents come from Grade 11 Functions (MCR3U), unit Exponential Functions. You will need a scientific calculator; a free graphing tool such as Desmos or GeoGebra is helpful but never required. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: Derivatives of Exponential and Sinusoidal Functions extends the rules to eˣ, aˣ, ln x, sin x and cos x. Later, Curve Sketching: Connecting f, f′ and f″ and Applications of Derivatives use the derivative to sketch graphs, describe motion and solve optimization problems.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Calculus and Vectors, Grade 12, University Preparation (MCV4U), strand A. Rate of Change: overall expectation A2 (the derivative function), specific expectations A2.1 to A2.3, and overall expectation A3 (properties of derivatives), specific expectations A3.1 to A3.5, for polynomial, rational and radical functions. Expectations A2.4 to A2.8 and the exponential and sinusoidal parts of A3.5 are taught in Derivatives of Exponential and Sinusoidal Functions; A1 is taught in Limits and Rates of Change. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will extend calculus beyond polynomials to the two function families that describe so much of the real world: exponential functions (growth, decay, cooling, interest) and sinusoidal functions (waves, tides, rotation, daylight).

The first half is about exponentials. You will discover by investigation that the derivative of aˣ is a constant multiple of aˣ, hunt down the special base e ≈ 2.71828 whose exponential function is its own derivative, meet the natural logarithm ln x as the inverse of eˣ, and prove that d/dx aˣ = aˣ ln a. Then you will combine these rules with the chain, product and quotient rules.

The second half is about sine and cosine. You will build tables and graphs of tangent slopes to find that d/dx sin x = cos x and d/dx cos x = −sin x (in radians), see why, and use the four-step derivative cycle. You will differentiate transformed waves, tan x and combinations such as x sin x and e^(−x) sin x, and finish by solving rate problems about populations, medication, cooling tea, Ferris wheels, Bay of Fundy tides, daylight and household electricity.

Remember from MHF4U: this unit builds on Advanced Functions: Exponential and Logarithmic Functions (logarithms and their laws), Radian Measure and Trigonometric Functions (radians and sinusoidal graphs), Trigonometric Identities and Equations (compound angle formulas) and Rates of Change (estimating instantaneous rates). From this course, you need the power, product, quotient and chain rules from Derivatives and Differentiation Rules. Keep a scientific calculator handy and set it to radian mode for every trig question. Graphing technology (a graphing calculator or free online graphing tool) is helpful for the investigations, but every answer can be reached by hand or with a scientific calculator.

Coming next: Curve Sketching: Connecting f, f′ and f″ uses these derivatives to analyse graphs, and Applications of Derivatives uses them for motion and optimization.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Calculus and Vectors, Grade 12, University Preparation (MCV4U), strand A. Rate of Change: specific expectations A2.4 to A2.8 (overall expectation A2, graphing derivatives of sinusoidal and exponential functions) and A3.5 (overall expectation A3, applying the product and chain rules to polynomial, sinusoidal, exponential, rational and radical functions). Expectations A2.1 to A2.3 and A3.1 to A3.4 are taught in Derivatives and Differentiation Rules, which also introduces A3.5 with polynomial, rational and radical functions. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In the units Derivatives and Differentiation Rules and Derivatives of Exponential and Sinusoidal Functions you learned to find derivatives. In this unit you learn what derivatives tell you about a graph. The first derivative f′ tells you where a function is increasing or decreasing and where its local maximums and minimums are. The second derivative f″ is the rate of change of the rate of change: it tells you how the graph bends (its concavity) and where it changes bend (points of inflection).

You will sketch f′ and f″ from a graph of f, read the shape of f from graphs of its derivatives, use sign charts, the first derivative test and the second derivative test, and describe the possible graphs that fit given derivative information. Then you put it all together to sketch polynomial functions from their equations by hand, and finish with rational functions, where vertical, horizontal and oblique asymptotes join the derivative information.

This is the toolkit that the next unit, Applications of Derivatives (motion, related rates and optimization), relies on. It builds on your MHF4U units Polynomial Functions and Equations and Rational Functions and Equations, where you sketched these graphs from zeros and asymptotes alone. Now calculus lets you place every turning point and every bend exactly. These ideas lead straight into first-year university calculus, economics (marginal cost and diminishing returns) and engineering.

In this unit derivatives stop being just slopes and start doing jobs. You will describe motion with position, velocity and acceleration (velocity is the first derivative of position and acceleration is the second), decide when an object is speeding up or slowing down, find total distance travelled, and move between position-time, velocity-time and acceleration-time graphs. You will read derivatives as real-world rates with units (inflation and deflation, population growth, flow rates), find instantaneous rates in exponential, sinusoidal and rational models (growth, radioactive decay, tides, drug concentration), use marginal cost, revenue and profit, and link changing quantities with the chain rule in related rates problems.

The second half is optimization: finding the absolute maximum or minimum of a quantity. You will learn the closed-interval method, then solve problems about areas and volumes, least material and cost, the best price for a bus fare or a concert, the fastest route for a canoe trip, and the best time to harvest timber, using polynomial, simple rational and exponential models. The unit ends with the full modelling cycle, built around the curriculum's own example of a foraging bird deciding how long to stay in a patch of food, and a model built from real-looking sales data.

Remember: this unit uses the derivative rules from Derivatives and Differentiation Rules and Derivatives of Exponential and Sinusoidal Functions, and the sign charts, derivative tests and concavity from Curve Sketching: Connecting f, f′ and f″. From Grade 11 Physics (SPH3U Kinematics) you may already know that velocity is the slope of a position-time graph; from MHF4U (Rates of Change) you know average and instantaneous rates of change. You will need a scientific calculator (radian mode for sine and cosine). Every lesson has worked examples, a hands-on activity, plenty of practice with full answers, and a short quiz. These ideas lead straight into first-year university calculus, physics, economics, engineering and the life sciences.

In this unit you meet vectors: quantities with both a magnitude and a direction, such as displacement, velocity and force. You will draw vectors as directed line segments, name directions as standard angles, true bearings (such as 320°) and quadrant bearings (such as N 40° W), and write vectors algebraically in Cartesian form [x, y] and polar form (r, θ). Trigonometry lets you convert between these forms in any quadrant.

Next you will add, subtract and multiply vectors by scalars, first geometrically with the triangle and parallelogram laws (and the cosine and sine laws), then with components. You will step into three-space, plot points (x, y, z), find distances and magnitudes, and work with 𝐢, 𝐣 and 𝐤. You will investigate and prove the properties of the operations, and use everything to solve real problems: resultants and equilibrants of forces, tensions in cables, airplanes in a crosswind, canoes crossing a river, and geometric proofs such as the midsegment theorem.

Notation: on this site vectors are printed in bold, like 𝐮 or 𝐀𝐁 (the vector from A to B). In your notebook, draw a small arrow over the letters instead. Vectors in components use square brackets, [3, −4], and points use round brackets, (3, −4).

Remember from earlier courses: the sine and cosine laws, bearings and angles in standard position from Grade 11 Functions (MCR3U, unit Trigonometric Ratios, Identities and Laws), and radians and cofunction identities from Advanced Functions (MHF4U, units Radian Measure and Trigonometric Functions and Trigonometric Identities and Equations). If you took Grade 11 Physics (SPH3U), the Kinematics and Forces units used vectors for motion and forces; this unit makes that work precise and extends it to three dimensions. You need a scientific calculator in degree mode.

Coming next: the unit Dot and Cross Products introduces two ways to multiply vectors (for angles, projections, work, area and torque), and Lines and Planes uses vectors to describe lines and planes in three-space.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Calculus and Vectors, Grade 12, University Preparation (MCV4U), strand C. Geometry and Algebra of Vectors: overall expectation C1 (representing vectors geometrically and algebraically, specific expectations C1.1 to C1.4) and part of C2 (operating with vectors, specific expectations C2.1 to C2.3). Expectations C2.4 to C2.8 are taught in Dot and Cross Products, and C3 and C4 in Lines and Planes. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will learn two new ways to multiply vectors, and use them to answer questions about angles, lengths, areas, volumes and forces.

The dot product u · v = |u||v| cos θ takes two vectors and gives a number. You will compute it geometrically and from components (u₁v₁ + u₂v₂ + u₃v₃), prove its properties, and use it to find the angle between two vectors, test for right angles, find direction angles, split a vector into projections along and across a direction, and calculate the work done by a force.

The cross product u × v of two vectors in three-space gives a vector perpendicular to both of them, chosen by the right-hand rule. Its length |u||v| sin θ is the area of a parallelogram. You will investigate its unusual properties (order matters, and grouping matters), and use it for areas of triangles and parallelograms, parallel tests and torque. Combining both products gives the triple scalar product, which measures the volume of a parallelepiped and tests whether vectors are coplanar. The last lesson brings everything together in problems about aircraft, runways, ramps, solar panels and more.

Remember from Vectors in Two-Space and Three-Space: vectors in Cartesian form, magnitude, vector addition, subtraction and scalar multiplication, and ground-velocity problems. From MCR3U (Trigonometric Ratios, Identities and Laws) you need the cosine law and the exact values of the special angles. From Grade 11 Physics (SPH3U, Work, Energy and Power) you may remember W = FΔd cos θ. You will need a scientific calculator in degree mode. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: in Lines and Planes you will use dot products and cross products to write equations of lines and planes, find normal vectors, and solve intersection and distance problems.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Calculus and Vectors, Grade 12, University Preparation (MCV4U), strand C. Geometry and Algebra of Vectors, overall expectation C2 (operating with vectors): specific expectations C2.4 to C2.8. Expectations C2.1 to C2.3 are taught in Vectors in Two-Space and Three-Space. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this final unit of Calculus and Vectors you will use vectors to describe lines and planes and to work out exactly how they meet. You will start in two-space, where a linear equation Ax + By + C = 0 is a line, and learn to write the same line with vector, parametric and symmetric equations. Then you will move to three-space, where lines need vector or parametric equations, and a single linear equation Ax + By + Cz + D = 0 is a plane whose coefficients form a normal vector. You will write planes in scalar, vector and parametric form, find them from three points with a cross product, and switch between forms.

The second half is about intersections and distances. You will find the line where two planes meet, sort pairs of lines into intersecting, parallel and skew, find where a line pierces a plane, picture all eight arrangements of three planes, and solve systems of three equations in three unknowns by elimination, connecting each algebraic result to its picture. You will finish by finding distances from a point to a line or plane and between skew lines.

Remember from earlier courses: linear systems in two variables and the distance from a point to a line (Grade 10, MPM2D Analytic Geometry), and vectors, the dot product and the cross product (the units Vectors in Two-Space and Three-Space and Dot and Cross Products of this course). Every lesson before the review has worked examples, a hands-on activity with things from home, and plenty of practice with full solutions. These ideas lead straight into first-year university linear algebra.