Calculus Amp; Vectors (mcv4u)
Ling Baus
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.
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.
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.
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.
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 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.
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.
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.
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.
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 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.
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 course builds on Advanced Functions to prepare you for university math. The two halves, (differential) calculus and vectors, are very different. Calculus is more algebraic and abstract, while vectors is more visual and geometric.
Chapter 7 test YouTube.pdf This is the solution manual for calculus and vectors. It comes in really handy when you want to check HOW a solution was arrived at. Thanks to MSMAMATH for uploading it in its entirety!
MCV4UAMMAR27.pdf Intersections of 3 planes (note: there was a handout today as well. Please see me for your very own copy of a note that would have taken you all period to write!) SMART Notebook.pdf from March 28,, 2018
The unit test for Chapter 9 will be on April 4th. We will have review days for the final evaluation on Thursday next week (March 29) and the second review day will be April 5th Final Evaluation April 6th
This course builds on students' previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.
Below is a course outline for the MCV4U course. The curriculum and outcomes remain the same whether you take this course in person or online. The curriculum is set by the Ministry of Education in Ontario, ensuring that all students follow the same requirements and gain the same skills, regardless of whether they complete the course in person or online.
MCV4U is intended for students interested in pursuing careers in fields such as economics, engineering, and science as well as some areas of business, including students who will need to take a university-level course in subjects such as algebra, calculus, or physics.
In this unit, students will be studying a quantity known as a vector. It is simply your typical line segment with a direction, but what is surprising is the number of applications that make use of it. For example, in the field of physics there is position, displacement, velocity, acceleration, forces, momentum, fields and so on. Students will look at some vector applications and solve related problems both geometrically and algebraically by performing operations on vectors.
In this unit, students will notice a shift in content as we move from the vectors portion of the course to the calculus portion. Students will be introduced to the idea of a rate of change. This is one of the most important concepts needed to understand calculus and will be used often throughout the course.
In this unit, students will develop the skills needed to sketch any given function. This unit will require students to connect algebraic concepts with graphical concepts in order to deepen their understanding of what the graph of a given function looks like.
What's up everybody, my name is Patrick and welcome to my page for MCV4U Calculus and Vectors. Click the Enroll button above to access free content. Scroll down to find testimonials and videos for the course organized by chapter. If you have any questions, please text/call me directly at 647-961-4348 or email [email protected]. If you're a student, I'll get back to you within a few hours.