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Master Separable Equations: Step-by-Step Solutions & Examples

Grade 12 Math: Master Advanced Concepts & Skills

12CV.A1.1

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Difference quotient: applications of functions

12CV.A1.2

Slope and equation of tangent line

Linear approximation

12CV.A1.3

Estimating Derivatives from a table

Critical number & maximum and minimum values

12CV.A1.4

Finding limits from graphs

Introduction to Calculus - Limits

12CV.A1.5

Definition of derivative

Limit laws

12CV.A1.6

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

12CV.A2.1

Intermediate value theorem

Curve sketching

12CV.A2.2

Power rule

Derivative of trigonometric functions

12CV.A2.3

Derivative of exponential functions

Derivative of logarithmic functions

12CV.A2.4

Implicit differentiation

Higher order derivatives

12CV.A2.5

Product rule

Quotient rule

12CV.A2.6

Chain rule

l'Hospital's rule

12CV.A2.7

Natural log: ln

Evaluating logarithms using change-of-base formula

Evaluating logarithms using logarithm rules

12CV.A2.8

Derivative of inverse trigonometric functions

Mean value theorem

12CV.A3.1

Position velocity acceleration

Related rates

12CV.A3.2

Optimization

Demand, revenue, cost & profit

12CV.A3.3

Marginal revenue, and maximizing revenue & average revenue

Marginal cost, and minimizing cost & average cost

12CV.A3.4

Marginal profit, and maximizing profit & average profit

Elasticity of demand

12CV.A3.5

Consumer and producer surplus

Continuous money flow

12CV.B1.1

Antiderivatives

Riemann sum

12CV.B1.2

Fundamental theorem of calculus

Definite integral

12CV.B1.3

U-Substitution

Integration using trigonometric identities

12CV.B1.4

Trigonometric substitution

Integration by parts

12CV.B1.5

Numerical integration

Integration of rational functions by partial fractions

12CV.B2.1

Areas between curves

Average value of a function

12CV.B2.2

Volumes of solids of revolution - Disc method

Volumes of solid with known cross-sections

12CV.B2.3

Volumes of solid of revolution - Shell method

Arc length

12CV.B2.4

Order and solutions to differential equations

Separable equations

12CV.B2.5

Modeling with differential equations

Slope fields

12CV.C1.1

Introduction to vectors

Magnitude of a vector

12CV.C1.2

Direction angle of a vector

Scalar multiplication of vectors

12CV.C1.3

Equivalent vectors

Adding and subtracting vectors in component form

12CV.C1.4

Operations on vectors in magnitude and direction form

Unit vector

12CV.C2.1

Word problems on vectors

Notation of matrices

12CV.C2.2

Adding and subtracting matrices

Scalar multiplication of matrices

12CV.C2.3

Matrix multiplication

The three types of matrix row operations

12CV.C2.4

Representing a linear system as a matrix

Solving a linear system with matrices using Gaussian elimination

12CV.C2.5

Zero matrix

Identity matrix

12CV.C2.6

Properties of matrix addition

Properties of scalar multiplication

12CV.C2.7

Properties of matrix multiplication

The determinant of a 2 x 2 matrix

12CV.C2.8

The determinant of a 3 x 3 matrix (General & Shortcut Method)

The Inverse of a 2 x 2 matrix

12CV.C3.1

Graphing linear inequalities in two variables

The inverse of 3 x 3 matrices with matrix row operations

The inverse of 3 x 3 matrix with determinants and adjugate

12CV.C3.2

2 x 2 invertible matrix

Solving linear systems using Cramer's Rule

12CV.C3.3

Solving linear systems using 2 x 2 inverse matrices

Transforming vectors with matrices

12CV.C4.1

General form:  Ax  + By  + C = 0

Point-slope form: y - y_1 = m(x - x_1)

Transforming shapes with matrices

Finding the transformation matrix

12CV.C4.2

3-D coordinate system

3-Dimensional vectors

12CV.C4.3

Dot product

Cross product

12CV.C4.4

3-Dimensional lines

3-Dimensional planes

12CV.C4.5

Cylindrical coordinates

Spherical-coordinates

12CV.C4.6

Calculus for vector functions

Tangent, normal, and binormal vectors

12CV.C4.7

Arc length with vector functions

Curvature with vector functions

Assignments

Challenge Zone

Foundation Review

We&apos;ve got you covered with our complete help for Grade 12 math, whether it&apos;s Ontario curriculum, <a href="https://www.alberta.ca/writing-diploma-exams.aspx">Diploma exam</a> (Alberta), <a href="http://www.edu.gov.mb.ca/k12/assess/tests/index.html">Provincial test</a> (Manitoba) or <a href="https://www.saskatchewan.ca/residents/education-and-learning/departmental-exams">Departmental exam</a> (Saskatchewan).

Aligned with your class or textbook, you will get grade 12 math help on topics like Trigonometric identities, Vectors, Permutations and Combinations, Matrices, Polynomial functions and so many more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest grade 12 math questions. Then, strengthen your understanding with tons of grade 12 math practice. 

All our lessons are taught by experienced Grade 12 math teachers. Let&apos;s finish your homework in no time, and ACE that final.

Grade 12 Calculus and Vectors, University Preparation (MCV4U)

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Overview

Mastering Separable Differential Equations: A Comprehensive Guide

What You'll Learn

Identify separable differential equations and rearrange them into separated form

Separate variables by moving all y terms to one side and all x terms to the other

Integrate both sides of a separated equation to find the general solution

Apply initial conditions to determine the constant and find particular solutions

Use algebraic techniques like partial fractions to integrate complex expressions

Isolate y when possible to express solutions in explicit form

What You'll Practice

Separating variables in differential equations with polynomial and exponential terms

Integrating expressions involving trigonometric functions, logarithms, and exponentials

Applying partial fraction decomposition to integrate rational expressions

Finding particular solutions using given initial conditions

Verifying solutions by checking algebraic steps and integration results

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

8 Video lessons

Practice exercises

Learning resources

Skills

Separable Equations

Variable Separation

Integration

Differential Equations

Initial Conditions

Particular Solutions

Partial Fractions

General Solutions