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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

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

12CV.A1.3

Estimating Derivatives from a table

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

12CV.A1.4

Finding limits from graphs

Introduction to Calculus - Limits

12CV.A1.5

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

12CV.A1.6

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

12CV.A2.1

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Curve sketching

12CV.A2.2

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

12CV.A2.3

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

12CV.A2.4

Implicit differentiation

Higher order derivatives

Higher Order Derivatives

First and second derivatives

First and Second Derivatives

Second derivatives with implicit differentiation

Implicit Differentiation

Higher Order Derivatives with Repeating Patterns

12CV.A2.5

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

12CV.A2.6

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

l'Hospital's rule

l'Hopital's rule

l'Hospital's Rule

Evaluating the limit using L'Hospital's 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

Mean Value Theorem

Mean Value Theorem Application

12CV.A3.1

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and 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

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

12CV.B1.2

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Fundamental Theorem of Calculus

Definite integral

Definite Integral

12CV.B1.3

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Integration using trigonometric identities

12CV.B1.4

Trigonometric substitution

Trigonometric Substitution

Integration by parts

12CV.B1.5

Numerical integration

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

Separable Equations

12CV.B2.5

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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