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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 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.
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.
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.