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# Calculus and Vectors

Students will extend their understanding of rates of change to include the derivatives of polynomial, rational, exponential, logarithmic, and trigonometric functions; and they will apply these to the modelling of real-world relationships. Integral calculus and its applications will be introduced. Students will solve problems involving vectors and lines and planes in three-space. This courseware is intended for students who have studied or are currently studying the Advanced Functions and Pre-Calculus courseware; will be required to take a university-level calculus, linear algebra or physics course; or may be considering the pursuit of studies in fields such as mathematics, computer science, engineering, science, business, or economics.

### Units

## Functions, Graphs, and Limits

In this unit, students will examine values of the average rate of change over an interval to approximate the instantaneous rate of change at a point. The concept of a limit will be formally defined, and students will use a graph of a function and the properties of limits to evaluate limits of a variety of functions.

The concept of a limit, as an approached value, will be reinforced by examining how Greek mathematicians developed the formula for the area of a circle. The two fundamental problems of calculus will be defined. Students will use the concept of a limit along with the average rate of change to approximate the instantaneous rate of change of a function at a point.

Students will learn the formal definition of a limit and the three conditions required for a limit to exist. Students will evaluate the limit of various functions at a particular value of \(x\) by observing the \(y\)-value(s) on a graph that are approached from the left and right side.

Students will learn 7 properties of limits and will apply these properties to algebraically evaluate limits of various functions.

In past explorations of functions and their graphs, students will have noticed that from start to end the graphs of some functions are made of one unbroken curve, whereas others include breaks within their domain. This module will use limits to define the three conditions that must be met for a function to be continuous throughout its domain. Also, students will learn the various types of discontinuities and the algebraic method of finding the location of a discontinuity.

Particular focus will be given to evaluating limits of polynomial and rational functions. Students will identify a rational function's removable discontinuity before simplifying the expression, and then apply limit properties to evaluate the limit.

Particular focus will be given to evaluating limits of functions containing radicals. Students will recall methods of rationalizing numerators and denominators as well as the domain and range of radical functions. Students will use the domain of a function to identify whether the limit exists before applying rationalization to evaluate limits of functions containing radicals.

Convergent and divergent sequences will be defined and students will observe large values of these sequences to determine if the limit exists at infinity. This module will connect limits at infinity with an algebraic method for determining the location of horizontal asymptotes.

## The Derivative

This unit will introduce the formal definition of the derivative. Students will examine graphs and use the definition of the derivative to verify the rules for determining derivatives: constant function rule, power rule, constant multiple rule, sum and difference rules, product rule, chain rule, and quotient rule. They will apply these rules to differentiate polynomial, rational, radical, and composite functions. Students will connect the value of the derivative at a particular value of *x* with the slope of the tangent line at a point on a curve, and they will use this slope and point to determine the equation of the tangent line.

Students will use the definition of the derivative to differentiate polynomial and rational functions as well as functions containing radicals. Through examining a variety of functions, students will identify the value(s)/interval(s) of \(x\) for which a function is not differentiable.

Through the examination of graphs, students will predict possible rules of differentiation, and then they will verify these rules by applying the definition of the derivative to general statements. The rules of differentiation explored in this module are the constant function rule, the power rule, the constant multiple rule, and the sum and difference rules.

Students will connect the value of the derivative at a specific value of \(x\) with the slope of the tangent line to a curve at a specific point. Students will use the value of the derivative with the slope-point equation of a line to determine the equation of a tangent line.

Students will use the definition of the derivative to develop the differentiation rule for the product of two functions. Students will then differentiate the product of two or more functions by applying the product rule.

Students will identify the inner and outer functions composing a composite function and then apply the chain rule to differentiate. Students will use the definition of the derivative to develop the chain rule

Students will develop the quotient rule by applying the product rule and chain rule to a quotient of two general functions. Students will then apply the quotient rule to differentiate rational functions and the quotient of two functions.

## Applications of Derivatives

In this unit, applications of the definition of the derivative are explored. We define higher order derivatives of a function, learn how to sketch the derivative of a function from the graph of the function, and see how instantaneous rates of change calculations can be used to solve real world problems in life sciences and the social sciences.

In this module, we will define the second derivative of a function, obtained by differentiating the derivative of the function, and more generally, the \(n^{th}\) derivative of a function for any positive integer \(n\).

In this module, we will learn how to sketch the graphs of the first and second derivatives of a polynomial function \(f\), given the graph of \(f\).

Applications of the derivative as an instantaneous rate of change are explored in the fields of geometry, life sciences, and the social sciences.

Applications of the first and second derivatives to problems involving motion are explored.

## Curve Sketching

In this unit, we develop an algorithm for sketching a curve given the algebraic equation of the curve. We discuss the extreme value and mean value theorems, and we examine the notion of a turning point, an absolute extreme, an interval of increase or decrease, concavity, and a point of inflection.

The notion of a turning point and an absolute extreme of a function are defined. We learn how to use the first derivative to locate turning points and the extreme values of a function over a closed interval.

The mean value theorem, which connects average rate of change and instantaneous rate of change, is stated and explored using examples.

In this module, we explore turning points of functions and introduce the first derivative test. More precisely, we will learn how to use the first derivative to find the intervals of increase and decrease of a given function.

The sign of the second derivative of a function can give information about the shape of the graph. The terms concave upward, concave downward, and point of inflection are defined, and the second derivative test is introduced.

Using the tools acquired throughout our study of functions, we develop an algorithm for sketching a curve given the equation of the curve. The functions studied include polynomials, rational functions, and functions involving radicals.

## Optimization and Related Rates

Now that we are familiar with how to calculate derivatives, we will use them in this unit to solve real-world problems in optimization and also as a way to determine related rates. We will also introduce Newton’s method as a way to approximate roots of equations.

We will express a given problem in mathematical language by determining a function that is to be maximized or minimized. Finding extreme values of this function allows us to solve problems such as maximizing area or minimizing time.

The problems in this module will be similar to those of the previous module, with the change of focus to maximizing revenue and profit or to minimizing costs. Marginal costs, marginal profits, and the demand function will be studied.

Some curves may be described by a relation such as \( x^2+y^2=9 \) where \( y \) is not given explicitly in terms of \( x \). In these cases, the method of implicit differentiation may be used to determine the derivative of one variable with respect to another.

In many real-world situations, a change in one quantity causes a change in another quantity or occurs together with a change in another quantity. That is, the two *rates* of change are *related*.

Linearization is the method of using a tangent line to approximate the function it is tangent to near the point of tangency. This module will also introduce an algorithm called Newton’s method for finding approximate roots of equations.

## Derivatives of Exponential, Logarithmic, and Trigonometric Functions

This unit begins with an introduction to Euler’s number, e. In addition to developing the derivatives of the exponential, logarithmic, and trigonometric functions, we will also extend our algebraic and equation solving skills with these three function types.

We will define \(e\) to equal the value of a fundamental limit. The relationship between \( f(x)=e^x \) and other exponential functions and its connection to the natural logarithm will be discussed. We will also investigate the derivative of \( f(x)=e^x \).

We will use a Maple investigation to help establish the derivative of \( f(x)=\ln(x) \) and then use our derivative rules to differentiate more complex functions involving the natural logarithm.

In this module, we will develop the derivative of any exponential function having a positive constant base.

In this module, we will develop the derivative of any logarithmic function having a positive constant base.

Through investigation, we develop the derivatives of both the sine and cosine functions, and then using two fundamental trigonometric limits and an identity, we prove our conjectures.

In this module, we develop and use the derivatives of each of the functions: \( f(x)=\tan(x)\), \(f(x)=\csc(x)\), \(f(x)=\sec(x)\), and \( f(x)=\cot(x) \).

## Applications of Exponential, Logarithmic, and Trigonometric Functions

In this unit, various applications of the derivatives of exponential, logarithmic, and trigonometric functions are explored. Familiar topics, including rates of change, curve sketching, optimization and related rates, will be revisited.

We have seen how calculus, more specifically the derivative, can be used to study rates of change of physical quantities. In this module, we will explore such applications, where the modeling equations involve exponential, logarithmic, and trigonometric functions.

L’Hospital’s rule is a tool for evaluating limits of indeterminate quotients that cannot be evaluated using the limit laws. This rule is particularly useful for evaluating limits of quotients involving exponential and logarithmic functions.

In this module, we will revisit the algorithm for curve sketching and apply it to curves whose equations involve exponential, logarithmic, and trigonometric functions.

Exponential, logarithmic, and trigonometric functions arise naturally in many real-world applications of calculus. We revisit the topic of optimization with a special focus on problems that involve functions of this sort.

In this module, we extend our study of related rates. In particular, we study the rates of change of quantities that are related via formulas involving exponential, logarithmic, or trigonometric functions.

## Integral Calculus

This unit introduces the second branch of calculus, called integral calculus, that is used for finding areas. The notion of an antiderivative, from differential calculus, and the definite integral are defined and connected using the fundamental theorem of calculus. The indefinite integral is introduced and methods for simplifying the process of integration are explored including: integration rules arising from known differentiation rules, helpful properties of integrals, the method of substitution, and integration by parts.

This module defines the notion of an antiderivative of a function and explores antiderivatives of integer powers of \(x\).

This module introduces the problem of calculating the total distance traveled over a time period when the velocity varies. This leads to the question of estimating areas of regions in the plane using rectangular approximations.

Sigma notation is a compact way of writing large sums of like terms. Riemann sums will be defined, using this notation, as a method for estimating net areas of regions in the plane.

The definite integral of a given function over a given interval is defined as the limit of Riemann sums. This module will introduce the process and terminology of integration.

In this module, we will explore examples where definite integrals can be evaluated using a net area interpretation, without considering Riemann sums.

This module presents some basic properties of definite integrals that will aid in simplifying the process of integration. The properties include order of integration, the zero rule, additivity, the constant multiple rule, and the sum and difference rules. The module ends with an exploration into the fundamental theorem of calculus.

The fundamental theorem of calculus connects the two branches of calculus: differential calculus and integral calculus. As a result of this theorem, we will gain a powerful tool for evaluating definite integrals using antiderivatives, without considering Riemann sums or net areas.

This module explores the antiderivatives of many familiar functions and defines the indefinite integral of a function. We will see that every differentiation rule gives rise to a corresponding rule for indefinite integration.

In this module, we evaluate definite integrals using a table of known indefinite integrals in conjunction with the fundamental theorem.

This module introduces one of the two main methods of integration: the method of substitution. This method arises from the chain rule for differentiation and allows us to simplify integrands using a change of variables.

This module introduces the second main method of integration: integration by parts. This method is derived from the product rule for differentiation and allows us to pass from the integrand in question to a new, hopefully simpler, integrand.

## Applications of Integral Calculus

In this unit, we will explore some applications of integral calculus. We will use definite integrals to calculate the net change of a quantity, volumes of three-dimensional solids, average values of functions, and lengths of curves. The end of this unit is devoted to the topic of differential equations, including a discussion of direction fields, solution sketching, separable equations and exponential growth and decay.

In this module, the fundamental theorem of calculus is reformulated in terms of net change. We will use this result to solve problems involving distance and displacement.

In this module, we use definite integrals to calculate the area of regions bounded by continuous curves.

In this module, we define the average value of a function and the length of a curve over a closed interval. We will see that definite integrals are central to the calculation of each of these quantities.

Very often, mathematical modelling results in the study of an equation involving the rate of change of a quantity. This is called a differential equation. We explore some well-known problems of this nature and introduce the necessary terminology for this topic.

It is often impossible to find an explicit formula for a solution to a particular differential equation. In this module, we learn how to sketch solutions to a differential equation without actually solving the given equation, and we use these sketches to obtain quantitative information about the solutions.

Most differential equations require graphical or numerical approaches when solving. In this module, we explore a certain family of differential equations — called separable equations — that can often be solved, explicitly, using indefinite integration.

Many physical quantities increase or decrease at a rate proportional to the amount of the quantity that is present. This property is known as the law of natural growth. In this module, we examine the family of differential equations, and their solutions, that arise in this context.

## Introduction to Vectors

This unit introduces the concept of a vector as being a mathematical object having both magnitude and direction. The mathematical operations on geometric vectors developed will culminate in the modeling and solving of problems involving the physical quantities of force and velocity.

This module introduces the representation of a vector as a directed line segment. The concepts of equal vectors, opposite vectors, angle between vectors, scalar multiplication, and unit vectors will be taught.

This module investigates the properties of vector addition, subtraction, and scalar multiplication. The triangle law and the parallelogram law will be taught.

How much force is required to pull a wagon? What is equilibrium and when do forces act on an object to produce it?

How does wind affect an airplane’s speed and direction? What other velocity problems can be solved by using vectors?

## Algebraic Vectors and Applications

This unit introduces vectors in a Cartesian coordinate system. The new model allows us to perform operations on vectors and to investigate interesting geometrical and physical applications.

This module connects the geometric model of a vector to the algebraic model. Components, position vectors, the 3-dimensional model, and direction angles (or direction cosines) will be taught.

This module investigates the properties of vector addition, subtraction, and scalar multiplication of algebraic vectors.

We will extend our operations on vectors to include the dot product (or scalar product). Properties of the dot product will be taught and used to find the angle between two vectors.

We will extend our operations on vectors to include the cross product (or vector product). Properties of the cross product, the triple scalar product, the triple vector product, the right hand rule, and the magnitude of the cross product will be investigated.

Vector projections will be taught and used to solve geometric and physical applications. We will discuss the concepts of work and the triple scalar product, and we will apply their definitions.

## Equations and Intersections of Lines in R^{2} and R^{3}

This unit extends our knowledge of the equations of lines to new forms involving vectors. We will consider these lines in both two and three dimensions, as well as determine intersections of and distances between lines.

We will look at two new forms for the equation of a line in the plane and consider the role vectors play in these new descriptions.

While the scalar equation or Cartesian equation of a line in R2 should look familiar, we will consider the role of vectors, specifically the normal vector, in describing a line in this form.

We extend the parametric and vector equations of lines from two to three dimensions. The symmetric equations of a line in R3 will also be introduced.

Do two lines in the plane always intersect? How about two lines in three dimensions? We will consider possible cases for the intersections of lines in \( \mathbb{R}^2 \) and \( \mathbb{R}^3 \) and determine the point of intersection where it exists.

We will revisit the projection of one vector onto another to help us determine the distance between a point and a line in \( \mathbb{R}^2 \). Will this same approach allow us to find the distance between a point and a line in \( \mathbb{R}^3 \)?

## Equations and Intersections of Planes

This unit introduces the various forms of the equations of planes and extends our techniques for solving systems of linear equations (such as the equations of planes). Row operations on matrices will be introduced to help find such algebraic solutions, which will then be interpreted geometrically.

This module extends our knowledge of the equations of lines to the vector and parametric equations of planes in \( \mathbb{R}^3 \).

We will extend our knowledge of a normal vector to help describe the equation of a plane in scalar form. We will also derive a formula for the distance between a point and a plane in \( \mathbb{R}^3 \) and then use this work to help determine the distance between skew lines.

Will a given line and a plane in \( \mathbb{R}^3 \) always intersect one another? If they do, how can we determine where they intersect? How might we sketch a plane?

What are the possible ways that two planes in \( \mathbb{R}^3 \) can intersect? We will solve the typical “two equations in three unknowns” to determine if the two planes do intersect one another.

Are you able to draw the different ways that three planes can intersect? Algebraically, we will introduce the matrix, Gaussian elimination, and row echelon form as tools used to determine if and where planes intersect. An investigation will help connect the algebraic solution to the system of equations to the geometry of the planes.

This module extends our work with matrices to Gauss-Jordan elimination and reduced row echelon form. We continue to explore the algebra and geometry determined by the different ways that three planes may intersect.