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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Equations and Intersections of Lines in R2 and R3

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.

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.