MATH 365 C7 - Study Guide: Unit 1
Analytic geometry is a branch of mathematics that enables us to see the relationships between geometrical shapes and their
mathematical representations, using coordinate systems. We begin Lesson 1.1 by considering the familiar—the Cartesian
coordinate system—which describes space using a two-dimensional, rectangular coordinate system. Here, we examine how
curves within the plane can be represented using parametric equations, and then we examine how our understanding of calculus
is modified in this context.
In Lesson 1.2 we turn our attention to another coordinate system: the polar coordinate system. We study the properties of
geometrical shapes defined by polar coordinates, as well as the relationship between the polar coordinate system and the
Cartesian system.
We finish with Lesson 1.3 by studying geometrical shapes known as conic sections—their formulas, graphical representations
and properties.
When you have completed this lesson, you will be able to
represent curves with parametric equations.
generate a function or implicit function for a parameterized curve.
calculate tangents to parametric curves.
calculate areas under parametric curves.
calculate arc lengths of parametric curves.
calculate surface areas defined by rotation of a parametric curve about an axis.
Reading: Chapter 10, Section 10.1: “Curves Defined by Parametric Equations”
Explore It: Section 10.1, including the “Explore & Test” exercises
“Explore it: Parametric Curves”
“Explore it: Families of Cycloids”
Lab: Lab 10.1: “Parametric Equations”
1
MATH 365 C7 - Study Guide: Unit 1
Video: Section 10.2, “Calculus with Parametric Curves”
Reading: Chapter 10, Section 10.2: “Calculus with Parametric Curves”
Lab: Lab 10.2: “Parametric Calculus”
When you have completed this lesson, you will be able to
describe the relationship between the Cartesian coordinate system and the polar coordinate system.
identify curves represented by polar equations.
sketch curves in polar coordinates.
transform polar equations into rectangular equations.
transform rectangular equations into polar equations.
give the equations of lines tangent to polar curves.
evaluate the arc length of polar curves.
evaluate the areas of regions enclosed by polar curves.
Video: Section 10.3, “Relationships between Polar and Rectangular Coordinates”
Reading: Chapter 10, Section 10.3: “Polar Coordinates”
Explore It: Section 10.3, “Explore it: Polar Curves,” including the “Explore & Test” exercises
Lab: Lab 10.3: “Polar Coordinates”
Video: Section 10.4: “Areas in Polar Coordinates”
Reading: Chapter 10, Section 10.4: “Areas and Lengths in Polar Coordinates”
Lab: Lab 10.4: “Polar Areas and Lengths”
When you have completed this lesson, you will be able to
sketch conic curves.
establish the quadratic equation of a given conic section.
2
MATH 365 C7 - Study Guide: Unit 1
Reading: Chapter 10, Section 10.5: “Conic Sections”
Explore It: Section 10.5, “Explore it: Conic Sections,” including the “Explore & Test” exercises
Video: Section 10.5
“Equations and Graphs of Parabolas”
“Equations and Graphs of Ellipses”
“Equations and Graphs of Hyperbolas”
Lab: Lab 10.5: “Conic Sections”
Video: Section 10.6, “Conic Sections in Polar Coordinates”
Reading: Chapter 10, Section 10.6, “Conic Sections in Polar Coordinates”
Lab: Lab 10.6, “Polar Conic Sections”
Complete Assignment 1 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
Complete Quiz 1 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13309). It is worth 2.5% of your course grade.
© Athabasca University
3
MATH 365 C7 - Study Guide: Unit 2
In this unit, we extend our study of calculus into three-dimensional space, describing the equations of lines, planes and other
surfaces using three-dimensional coordinate systems. We begin in Lesson 2.1 by extending the concept of two-dimensional
space, as described by the 2D Cartesian coordinate system, to three dimensions, by introducing the 3D Cartesian coordinate
system.
In Lesson 2.2, we introduce the concept of vectors, studying their properties and operations, and then we examine some
applications of vectors in physics. In Lesson 2.3 we explore how vectors are used to define the equations of lines and planes in a
three-dimensional space in the form of both vector equations and parametric equations. In Lesson 2.4, we broaden our study of
surfaces beyond Lesson 2.1 to include cylinders and to extend the concept of two-dimensional conics to three-dimensional
quadric surfaces.
When you have completed this lesson, you will be able to
represent points in a three-dimensional space.
represent surfaces in a three-dimensional space.
identify and describe spheres from their equations in a three-dimensional space.
calculate the distance between points in a three-dimensional space.
Video: Section 12.1, “The Three-Dimensional Rectangular Coordinate System”
Reading: Chapter 12, Section 12.1: “Three-Dimensional Coordinate Systems”
Lab: Lab 12.1: “3D Coordinate Systems”
When you have completed this lesson, you will be able to
represent vectors geometrically in two- and three-dimensional spaces.
represent vectors using orthogonal components.
perform arithmetic operations with vectors.
determine the dot product of two vectors.
determine the cross product of two vectors.
1
MATH 365 C7 - Study Guide: Unit 2
determine the angle between vectors.
describe the geometric properties of the dot product and cross product.
Reading: Chapter 12, Section 12.2: “Vectors”
Explore It: Section 12.2, “Explore It: Adding Vectors,” including the “Explore & Test” exercises
Video: Section 12.2
“Geometric Description of Vectors”
“Combining Vectors in Space”
Lab: Lab 12.2, “Vectors”
Reading: Chapter 12, Section 12.3, “The Dot Product”
Explore It: Section 12.3, including the “Explore & Test” exercises
Explore It: The Dot Product of Two Vectors
Explore It: Vector Projections
Video: Section 12.3, “The Dot Product: Work”
Lab: Lab 12.3, “Dot Product”
Reading: Chapter 12, Section 12.4, “The Cross Product”
Explore It: Section 12.4, “Explore It: The Cross Product,” including the “Explore & Test” exercises
Video: Section 12.4, “Properties of Cross Product”
Lab: Lab 12.4, “Cross Product”
When you have completed this lesson, you will be able to
determine the vector and parametric equations of lines and line segments in 3D space.
establish the relation between a vector and a parametric equation of a line.
determine the vector and scalar equations of planes in 3D space.
find the distance between a point and a plane in 3D space.
Video: Section 12.5, “Equations of Lines in Space”
Reading: Chapter 12, Section 12.5, “Equations of Lines and Planes”
Lab: Lab 12.5, “Lines and Planes”
2
MATH 365 C7 - Study Guide: Unit 2
When you have completed this lesson, you will be able to
identify and sketch cylinders from their equations.
identify and sketch quadric surfaces from their equations.
classify types of quadric surfaces from their equations.
describe quadric surfaces and how they may be used in applications.
Reading: Chapter 12, Section 12.6, “Cylinders and Quadric Surfaces”
Explore It: Section 12.6, including the “Explore & Test” exercises
“Explore It: Traces of a Surface”
“Explore It: Quadric Surfaces”
Video: Section 12.6, “Cylinders”
Lab: Lab 12.6, “Quadric Surfaces”
Complete Assignment 2 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
© Athabasca University
3
MATH 365 C7 - Study Guide: Unit 3
In this unit, we expand on our previous discussion of vectors and three-dimensional space to consider the calculus of vectorvalued functions. So far, we have considered only real-valued functions. To describe curves and surfaces in three-dimensional
space, vector functions are needed. In Lesson 3.1, we introduce the concepts of vector functions and space curves, and we learn
how to differentiate and integrate vector functions. We examine familiar calculus topics, such as limits, continuity and tangent
lines, now within the context of vector functions.
In Lesson 3.2, we see how the calculus of vector functions can be used to determine the curvature and twisting of the curves and
the arc length along the curves. Vector functions are also used to describe the movement of particles through curves, as we shall
study in Lesson 3.3. To learn about the velocity and acceleration of these particles, differentiation of vector functions is required.
When you have completed this lesson, you will be able to
define a vector-valued function and space curve.
describe the meaning of calculus concepts in terms of vector-valued functions, including limits, continuity, differentiation
and integration.
describe and sketch curves defined by vector functions.
determine vector functions to describe given curves.
determine the parametric equations of a space curve.
apply the rules of differentiation to vector functions.
determine the tangent vector and a vector equation or parametric equations of a line tangent to a curve in 3D space.
evaluate definite integrals of vector functions.
Reading: Chapter 13, Section 13.1, “Vector Functions and Space Curves”
Explore It: Section 13.1, including the “Explore & Test” exercises
“Explore It: Vector-Valued Functions and Space Curves”
“Explore It: The Twisted Cubic Curve”
“Explore It: Visualizing Space Curves as the Intersection of Surface”
Video: Section 13.1, “Vector Functions and Space Curves”
Lab: Lab 13.1, “Vector Functions”
Reading: Chapter 13, Section 13.2, “Derivatives and Integrals of Vector Functions”
Explore It: Section 13.2, “Explore It: Secant and Tangent Vectors,” including the “Explore & Test” exercises
1
MATH 365 C7 - Study Guide: Unit 3
Video: Section 13.2, “Derivatives of Vector Functions”
Lab: Lab 13.2, “Calculus of Vectors”
When you have completed this lesson, you will be able to
find the arc length of a space curve.
obtain the parametrization of a vector-valued function or space curve.
define the arc length of a curve using a function.
parametrize a curve with respect to arc length.
determine the curvature of a vector-valued function or space curve.
determine the unit normal and binormal vectors of a vector-valued function.
Reading: Chapter 13, Section 13.3, “Arc Length and Curvature”
Explore It: Section 13.3, including the “Explore & Test” exercises
“Explore It: The Unit Tangent Vector”
“Explore It: The TNB Frame”
“Explore It: Osculating Circle”
Video: Section 13.3, “Arc Length of a Space Curve”
Lab: Lab 13.3, “Arc Length and Curvature”
When you have completed this lesson, you will be able to
find the position, velocity and acceleration vectors of a particle moving along a space curve.
apply the vector version of Newton’s Second Law of Motion.
use vector equations to describe the projectile motion of a particle.
determine the parametric equations of the trajectory of a particle in projectile motion.
determine the normal and tangential component of acceleration of a moving particle in a three-dimensional space.
use vector equations to understand the proofs of Kepler’s laws of planetary motion.
Reading: Chapter 13, Section 13.4, “Motion in Space: Velocity and Acceleration”
Explore It: Section 13.4, “Explore It: Velocity and Acceleration Vectors,” including the “Explore & Test” exercises
Video: Section 13.4, “Motion in Space: Velocity and Acceleration - Projectile Motion”
Lab: Lab 13.4, “Motion in Space”
2
MATH 365 C7 - Study Guide: Unit 3
Complete Assignment 3 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
Complete Quiz 2 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13309). It is worth 2.5% of your course grade.
Complete the midterm exam (https://science22.lms.athabascau.ca/mod/page/view.php?id=13243). It covers Units 1 to 3 of the
Study Guide (textbook chapters 10, 12 and 13). It is worth 25% of your course mark. The midterm exam is designed to take 1 to 3
hours to complete.
© Athabasca University
3
MATH 365 C7 - Study Guide: Unit 4
In this unit, we begin to extend the basic concepts of single variable calculus to multivariable functions. The concepts of calculus
as they apply to single variable functions extend naturally to functions of several variables; however, as multivariable functions
are intrinsically more complicated than functions of one variable, we need to develop new approaches and techniques to work
with such functions. We begin in Lesson 4.1 where we provide context for functions of several variables, and then study the
familiar concepts of limits and continuity in this context.
In Lesson 4.2, we consider differentiation in the context of multivariable functions, first by introducing the concept of the partial
derivative, and learn how partial derivatives enable us to describe tangent planes, directional derivatives and the gradient vector.
In Lesson 4.3 we shift our attention to applications of differentiation to solve optimization problems involving functions of
several variables and to discuss Lagrange multipliers.
When you have completed this lesson, you will be able to
graph functions of two variables.
describe the level curves of functions of two variables.
establish the relationship between limits and limits along smooth curves.
identify discontinuities of functions of two variables.
Video: Section 14.1, “Functions of Two Variables”
Reading: Chapter 14, Section 14.1, “Functions of Several Variables”
Explore It: Section 14.1, including the “Explore & Test” exercises
“Explore It: Animated Level Curves”
“Explore It: Level Curves of a Surface”
Lab: Lab 14.1, “Multivariate Functions”
Reading: Chapter 14, Section 14.2, “Limits and Continuity”
Explore It: Section 14.2, “Limit That Does Not Exist,” including the “Explore & Test” exercises
Video: Section 14.2, “Limits of Functions of Two Variables”
Lab: Lab 14.2, “Limits and Continuity”
1
MATH 365 C7 - Study Guide: Unit 4
When you have completed this lesson, you will be able to
calculate the partial derivative of multivariate functions.
calculate higher order partial derivatives of multivariate functions.
give the equation of a plane tangent to a surface.
use differentials to approximate the value of functions of several variables.
use local linear approximations to estimate errors in approximation.
apply the Chain Rule for derivatives of functions of several variables.
obtain directional derivatives of functions of two and three variables.
give the geometrical interpretation of the gradient of a function of two variables.
Reading: Chapter 14, Section 14.3, “Partial Derivatives”
Video: Section 14.3, “Partial Derivatives”
Lab: Lab 14.3, “Partial Derivatives”
Reading: Chapter 14, Section 14.4, “Tangent Planes and Linear Approximation”
Explore It: Section 14.4, “The Tangent Plane of a Surface,” including the “Explore & Test” exercises
Video: Section 14.4, “Tangent Planes”
Lab: Lab 14.4, “Tangent Planes”
Reading: Chapter 14, Section 14.5, “The Chain Rule.”
Video: Section 14.5, “Partial Derivatives: The Chain Rule”
Lab: Lab 14.5, “Chain Rule”
Reading: Chapter 14, Section 14.6, “Directional Derivatives and the Gradient Vector”
Explore It: Section 14.6, including the “Explore & Test” exercises
“Explore It: Directional Derivatives of a Surface in the First Octant”
“Explore It: Directional Derivatives”
“Explore It: Maximizing Directional Derivatives”
Lab: Lab 14.6, “Gradient”
When you have completed this lesson, you will be able to:
obtain the extreme values of functions of several variables.
obtain absolute extreme values of functions of several variables.
apply Lagrange multipliers to obtain the extreme values of functions of two and three variables.
2
MATH 365 C7 - Study Guide: Unit 4
Reading: Chapter 14, Section 14.7, “Maximum and Minimum Values”
Explore It: Section 14.7, including the “Explore & Test” exercises
“Explore It: Families of Surfaces”
“Explore It: Critical Points from Contour Maps”
Video: Section 14.7, “Partial Derivatives: Maximum and Minimum Values”
Lab: Lab 14.7, “Max. and Min. Values”
Reading: Chapter 14, Section 14.8, “Lagrange Multipliers”
Explore It: Section 14.8, “Lagrange Multipliers,” including the “Explore & Test” exercises
Lab: Lab 14.8, “Lagrange Multipliers”
Complete Assignment 4 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
© Athabasca University
3
MATH 365 C7 - Study Guide: Unit 5
In this unit, we continue our study of multivariable functions. In particular, we focus on extending the notion of the definite
integral from single variable functions to functions of several variables, through the introduction of multiple integrals. We begin
our exploration of multiple integrals in Lesson 5.1 by examining double integrals of functions of two variables, first over
rectangular regions of Euclidean space, and then over general regions within Euclidean space. We end by examining double
integrals taken over regions within the polar coordinate system. We study some applications along the way.
In Lesson 5.2 we continue our exploration of multiple integrals, shifting our attention to triple integrals of functions of three
variables. Again, we begin with the simplest case in which our triple integrals are taken over rectangular regions of Euclidean
space, and then over general regions within Euclidean space. We then observe that the introduction of cylindrical coordinates
and spherical coordinates greatly simplifies the computation of triple integrals over particular types of three-dimensional solid
regions. During our examination of multiple integrals, we observe that, perhaps, the most challenging task when considering
multiple integration is determining the correct setup of the integrals. The evaluation of the double and triple integrals might be
the easier part, but it is not necessarily simple. Therefore, in the final lesson, Lesson 5.3, we study how to use a change of
variables to simplify multiple integrals.
When you have completed this lesson, you will be able to
state the properties of double integrals.
integrate double integrals over rectangular regions.
state and apply Fubini’s Theorem for double integrals.
evaluate double integrals over general regions of type I and type II.
evaluate double integrals in polar coordinates.
Video: Section 15.1, “Double Integrals over Rectangles - Volume and Iterated Integrals”
Reading: Chapter 15, Section 15.1, “Double Integrals over Rectangles”
Explore It: Section 15.1, “Fubini’s Theorem,” including the “Explore & Test” exercises
Lab: Lab 15.1, “Double Integrals over Rectangles”
Video: Section 15.2, “Double Integrals over General Regions”
Reading: Chapter 15, Section 15.2, “Double Integrals over General Regions”
Lab: Lab 15.2, “General Double Integrals”
Video: Section 15.3, “Double Integrals in Polar Coordinates”
1
MATH 365 C7 - Study Guide: Unit 5
Reading: Chapter 15, Section 15.3, “Double Integrals in Polar Coordinates”
Lab: Lab 15.3, “Polar Double Integrals”
Reading: Chapter 15, Section 15.4, “Applications of Double Integrals”
Video: Section 15.4, “Applications of Double Integrals - Probability”
Lab: Lab 15.4, “Double Integral Applications”
Video: Section 15.5, “Surface Area”
Reading: Chapter 15, Section 15.5, “Surface Area”
Lab: Lab 15.5, “Surface Area”
When you have completed this lesson, you will be able to
evaluate triple integrals over rectangular boxes.
evaluate triple integrals over simple
-solids.
state and apply Fubini’s Theorem for triple integrals.
evaluate triple integrals in cylindrical and spherical coordinates.
Video: Section 15.6, “Triple Integrals”
Reading: Chapter 15, Section 15.6, “Triple Integrals”
Explore It: Section 15.6, “Explore It: Regions of Triple Integrals,” including the “Explore & Test” exercises
Lab: Lab 15.6, “Triple Integrals”
Reading: Chapter 15, Section 15.7, “Triple Integrals in Cylindrical Coordinates”
Video: Section 15.7, “Triple Integrals in Cylindrical Coordinates”
Lab: Lab 15.7, “Cylindrical Triple Ints”
Video: Section 15.8, “Triple Integrals in Spherical Coordinates”
Reading: Chapter 15, Section 15.8, “Triple Integrals in Spherical Coordinates”
Explore It: Section 15.8, including the “Explore & Test” exercises
“Explore it: Cylindrical and Spherical Coordinates”
“Explore it: Surfaces in Cylindrical and Spherical Coordinates”
“Explore it: A Region in Spherical Coordinates”
Lab: Lab 15.8, “Spherical Triple Ints”
When you have completed this lesson, you will be able to
2
MATH 365 C7 - Study Guide: Unit 5
convert triple integrals from rectangular to spherical and cylindrical coordinates.
evaluate double and triple integrals by changing variables through linear transformations.
Reading: Chapter 15, Section 15.9, “Change of Variables in Multiple Integrals”
Lab: Lab 15.9, “Change of Variables”
Complete Assignment 5 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
Complete Quiz 3 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13309). It is worth 2.5% of your course grade.
© Athabasca University
3
MATH 365 C7 - Study Guide: Unit 6
We end this course with the study of vector fields, which are functions that assign a vector to each point in space. Vector fields
provide a natural description of flows. For example, water or air currents can be described using vector fields, where a vector
specifies the direction and speed of water or air flow at each point in space. Vector fields are introduced in Lesson 6.1.
In Lesson 6.2 we introduce the notion of a line integral, which will help us study flows within vector fields. For example, the line
integral will enable us to calculate the work done by a variable force as it moves a particle along a pathway between points a and
b. In our exploration of line integrals, we will revisit the Fundamental Theorem of Calculus by way of the Fundamental Theorem
for Line Integrals and Green’s Theorem. In Lesson 6.3 we introduce two new operations, curl and divergence, which we will
study in the context of parametric surfaces and surface integrals (Lesson 6.4), culminating in Stokes’ Theorem and the
Divergence Theorem. Our final lesson, Lesson 6.4, highlights the relationship between the Fundamental Theorem of Calculus,
The Fundamental Theorem for Line integrals, Green’s Theorem, Stokes’ Theorem and the Divergence Theorem.
In Lesson 6.2 we introduce the notion of a line integral, which will help us study flows within vector fields. For example, the line
integral will enable us to calculate the work done by a variable force as it moves a particle along a pathway between points a and
b. In our exploration of line integrals, we will revisit the Fundamental Theorem of Calculus by way of the Fundamental Theorem
for Line Integrals and Green’s Theorem. In Lesson 6.3 we introduce two new operations, curl and divergence, which we will
study in the context of parametric surfaces and surface integrals (Lesson 6.4), culminating in Stokes’ Theorem and the
Divergence Theorem. Our final lesson, Lesson 6.4, highlights the relationship between the Fundamental Theorem of Calculus,
The Fundamental Theorem for Line integrals, Green’s Theorem, Stokes’ Theorem and the Divergence Theorem.
When you have completed this lesson, you will be able to
describe the geometrical representation of vector fields.
find the gradient vector field of a multivariable scalar function.
Video: Section 16.1, “Vector Fields”
Reading: Chapter 16, Section 16.1, “Vector Fields”
Explore It: Section 16.1, “Explore It: Vector Fields,” including the “Explore & Test” exercises
Lab: Lab 16.1, “Vector Fields”
When you have completed this lesson, you will be able to
1
MATH 365 C7 - Study Guide: Unit 6
evaluate line integrals along curves, piecewise smooth curves and closed paths.
establish when a line integral is independent of the path.
state and apply Green’s Theorem.
Reading: Chapter 16, Section 16.2, “Line Integrals”
Explore It: Section 16.2, “Explore It: Line Integrals,” including the “Explore & Test” exercises
Lab: Lab 16.2, “Line Integrals”
Video: Section 16.3, “The Fundamental Theorem of Line Integrals”
Reading: Chapter 16, Section 16.3, “The Fundamental Theorem for Line Integrals”
Explore It: Section 16.3, “Explore It: FTC for Line Integrals,” including the “Explore & Test” exercises
Lab: Lab 16.3, “FTC Line Integrals”
Video: Section 16.4, “Green’s Theorem”
Reading: Chapter 16, Section 16.4, “Green’s Theorem”
Explore It: Section 16.4, “Explore It: Green’s Theorem,” including the “Explore & Test” exercises
Lab: Lab 16.4, “Green’s Theorem”
When you have completed this lesson, you will be able to
establish the geometrical interpretation of the divergence and curl fields.
evaluate surface integrals.
state and apply Stokes’ Theorem.
state and apply the Divergence Theorem.
Video: Section 16.5, “Curl”
Reading: Chapter 16, Section 16.5, “Curl and Divergence”
Lab: Lab 16.5, “Curl and Divergence”
Video: Section 16.6, “Parametric Surfaces”
Reading: Chapter 16, Section 16.6, “Parametric Surfaces and Their Areas”
Explore It: Section 16.6, including the “Explore & Test” exercises
“Explore It: Grid Curves on Parametric Surfaces”
“Explore It: Families of Parametric Surfaces”
Lab: Lab 16.6, “Parametric Surfaces”
Video: Section 16.7, “Surface Integrals”
Reading: Chapter 16, Section 16.7, “Surface Integrals”
2
MATH 365 C7 - Study Guide: Unit 6
Explore It: Section 16.7, “Explore It: A Nonorientable Surface,” including the “Explore & Test” exercises
Lab: Lab 16.7, “Surface Integrals”
Video: Section 16.8, “Stokes’ Theorem”
Reading: Chapter 16, Section 16.8, “Stokes’ Theorem”
Lab: Lab 16.8, “Stokes’ Theorem”
Video: Section 16.9, “The Divergence Theorem”
Reading: Chapter 16, Section 16.9, “The Divergence Theorem”
Lab: Lab 16.9, “Divergence Theorem”
When you have completed this lesson, you will be able to
describe the relationship between The Green’s Theorem and Stokes’ Theorem.
Reading: Chapter 16, Section 16.10, “Summary”
Complete Assignment 6 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13210). It is worth 5% of your course
grade.
Complete Quiz 4 (https://science22.lms.athabascau.ca/mod/page/view.php?id=13309). It is worth 2.5% of your course grade.
Complete the final exam (https://science22.lms.athabascau.ca/mod/page/view.php?id=13243). It covers Units 4 to 6 of the
Study Guide (textbook chapters 14, 15 and 16). It is worth 35% of your course mark. The final exam is designed to take 1 to 3
hours to complete.
© Athabasca University
3