MA129

Laurier

Course Overview

Lessons & Practice

I. Welcome

1. Pre-Calculus (Review)

15hr

1.1.1. Terminology Review5 min

1.1.2. Practice

1.1.3. Multiplying by a Monomial7 min

1.1.4. Example3 min

1.1.5. Practice2 min

1.2.1. Binomial

\times

1.2.2. Using Algebra Tiles3 min

1.2.3. Example7 min

1.2.4. Practice Level 15 min

1.2.5. Practice Level 29 min

1.2.6. Practice Level 38 min

1.3.1. Binomial

\times

1.3.2. Example3 min

1.3.3. Practice Level 13 min

1.3.4. Practice Level 1

1.3.5. Practice Level 2

1.3.6. Practice Level 3

1.4.1. Factoring 5 min

1.4.2. Factoring a Difference & Sum of Cubes3 min

1.4.3. Example2 min

1.4.4. Practice46 sec

1.4.5. Practice2 min

1.4.6. Practice2 min

1.5.1. Factoring Summary3 min

1.5.2. Factoring Drills 💪 Level 210 min

1.5.3. Practice Level 33 min

1.5.4. Practice Level 34 min

1.5.5. Practice Level 35 min

1.5.6. Practice Level 34 min

1.6.1. Rational Expressions & Functions7 min

1.6.2. Practice Level 13 min

1.6.3. Practice Level 24 min

1.6.4. Practice Level 25 min

1.6.5. Practice Level 34 min

1.7.1. Multiplying & Dividing Rational Expressions6 min

1.7.2. Practice Level 13 min

1.7.3. Practice Level 26 min

1.7.4. Practice Level 35 min

1.8.1. Adding and Subtracting Rational Expressions5 min

1.8.2. Practice Level 14 min

1.8.3. Practice Level 1

1.8.4. Practice Level 25 min

1.8.5. Practice Level 34 min

1.9.1. Solving Equations2 min

1.9.2. Example8 min

1.9.3. Practice2 min

1.10.1. Solving Multi-Step Equations1 min

1.10.2. Example7 min

1.10.3. Practice Level 16 min

1.10.4. Example6 min

1.10.5. Practice Level 13 min

1.10.6. Example5 min

1.10.7. Practice Level 24 min

1.10.8. Example7 min

1.10.9. Practice Level 26 min

1.11.1. Notes About Solving Equations & Formulas

1.11.2. Practice Level 23 min

1.11.3. Practice Level 25 min

1.11.4. Practice Level 35 min

1.11.5. Practice Level 35 min

1.11.6. Practice Level 32 min

1.12.1. Interval Notation 4 min

1.12.2. Set Notation3 min

1.12.3. Example4 min

1.12.4. Practice2 min

1.12.5. Practice52 sec

1.12.6. Cheatsheet

1.13.1. Inequalities5 min

1.13.2. Practice2 min

1.13.3. Representing Inequalities8 min

1.13.4. Practice4 min

1.13.5. Practice3 min

1.14.1. Solving Inequalities2 min

1.14.2. Example (Single Step Inequalities)5 min

1.14.3. Practice Level 12 min

1.14.4. Practice Level 13 min

1.14.5. Example (Multi-Step Inequalities)3 min

1.14.6. Practice Level 22 min

1.14.7. Practice Level 33 min

1.14.8. Practice Level 33 min

1.15.1. Solving Compound Inequalities4 min

1.15.2. Example51 sec

1.15.3. Practice Level 22 min

1.16.1. Investigating Integer Exponents6 min

1.16.2. Integer Exponents1 min

1.16.3. Practice1 min

1.16.4. Practice4 min

1.16.5. Practice3 min

1.17.1. Exponent Rules4 min

1.18.1. Exponent Rules for Fractions & Products5 min

1.18.2. Example3 min

1.18.3. Practice Level 12 min

1.18.4. Practice Level 12 min

1.18.5. Practice Level 23 min

1.18.6. Practice Level 34 min

1.19.1. Exponent Product Rule for Multiplying Powers3 min

1.19.2. Example2 min

1.19.3. Practice Level 13 min

1.19.4. Practice Level 25 min

1.20.1. Exponent Quotient Rule for Dividing Powers3 min

1.20.2. Example2 min

1.20.3. Practice Level 14 min

1.20.4. Practice Level 24 min

1.21.1. Exponent Power of a Power Rule3 min

1.21.2. Example56 sec

1.21.3. Practice Level 14 min

1.21.4. Practice Level 24 min

1.21.5. Example (⭐Important!)4 min

1.21.6. Practice Level 34 min

1.22.1. Investigating Rational Exponents5 min

1.22.2. Rational Exponents2 min

1.22.3. Practice Level 19 min

1.22.4. Practice Level 24 min

1.22.5. Practice Level 34 min

1.23.1. Practice Level 15 min

1.23.2. Practice Level 23 min

1.23.3. Practice Level 37 min

1.24.1. Radical Expressions5 min

1.24.2. Multiplying Radicals6 min

1.24.3. Example2 min

1.24.4. Practice Level 12 min

1.24.5. Practice Level 25 min

1.24.6. Practice Level 34 min

1.25.1. Rationalizing the Denominator5 min

1.25.2. Simplifying Radicals7 min

1.25.3. Example4 min

1.25.4. Practice Level 12 min

1.25.5. Practice Level 25 min

1.25.6. Practice Level 33 min

1.26.1. Domain2 min

1.26.2. Range2 min

1.26.3. Example2 min

1.26.4. Practice 40 sec

1.26.5. Practice 3 min

1.27.1. Equation of a Line4 min

1.27.2. Example5 min

1.27.3. Practice4 min

1.27.4. Horizontal & Vertical Lines4 min

1.27.5. Practice2 min

1.28.1. Slope of a Line8 min

1.28.2. Practice Level 12 min

1.28.3. Finding the Slope of a Line8 min

1.28.4. Practice Level 24 min

1.28.5. Practice Level 36 min

1.28.6. Example4 min

1.28.7. Practice Level 33 min

1.29.1. Different Forms of the Equation of a Line4 min

1.29.2. Example9 min

1.29.3. Practice Level 14 min

1.29.4. Practice Level 23 min

1.29.5. Practice Level 213 min

1.29.6. Practice Level 34 min

1.30.1. Graphing Linear Equations - Table of Values3 min

1.30.2. Graphing Linear Equations - Slope y-Intercept Form5 min

1.30.3. Practice6 min

1.30.4. Graphing Linear Equations - Using Intercepts4 min

1.30.5. Practice5 min

1.30.6. Practice4 min

1.31.1. Parallel & Perpendicular Lines7 min

1.31.2. Practice Level 25 min

1.31.3. Practice Level 24 min

1.32.1. Finding the Equation of a Line4 min

1.32.2. Practice Level 27 min

1.32.3. Practice Level 35 min

1.33.1. Quadratic Relations13 min

1.33.2. Example7 min

1.33.3. Practice Level 15 min

1.33.4. Practice Level 25 min

1.33.5. Practice Level 37 min

1.34.1. Properties of Quadratic Graphs5 min

1.34.2. Example9 min

1.34.3. Practice Level 13 min

1.34.4. Practice Level 13 min

1.34.5. Practice Level 23 min

1.34.6. Practice Level 37 min

1.34.7. Practice Level 38 min

1.35.1. Factored Form of a Quadratic Equation9 min

1.35.2. Example8 min

1.35.3. Practice Level 14 min

1.35.4. Practice Level 27 min

1.35.5. Practice Level 35 min

1.35.6. Summary - Factored Form of a Quadratic Equation

1.36.1. Example7 min

1.36.2. Practice Level 16 min

1.36.3. Example5 min

1.36.4. Practice Level 211 min

1.36.5. Practice Level 34 min

1.36.6. Practice Level 3 🔥3 min

1.37.1. Vertical Stretches and Reflections7 min

1.37.2. Practice2 min

1.37.3. Practice2 min

1.37.4. Vertical & Horizontal Translations (Shifts)7 min

1.37.5. Example2 min

1.37.6. Practice2 min

1.37.7. Practice3 min

1.38.1. Graphing Quadratics in Vertex Form3 min

1.38.2. Example5 min

1.38.3. Practice Level 12 min

1.38.4. Practice Level 12 min

1.38.5. Practice Level 25 min

1.38.6. Practice Level 33 min

1.38.7. Practice Level 34 min

1.39.1. Example2 min

1.39.2. Practice Level 11 min

1.39.3. Practice Level 22 min

1.39.4. Practice Level 33 min

1.39.5. Practice Level 32 min

1.40.1. Characteristics of Exponential Functions2 min

1.40.2. Example53 sec

1.40.3. Practice Level 12 min

1.40.4. Practice Level 22 min

1.41.1. Transformations of Exponential Functions3 min

1.41.2. Example3 min

1.41.3. Practice Level 12 min

1.41.4. Practice Level 23 min

1.41.5. Practice Level 29 min

1.42.1. Solving Exponential Equations1 min

1.42.2. Example2 min

1.42.3. Practice Level 1

1.42.4. Practice Level 2

1.42.5. Practice Level 3

1.43.1. Applications of Exponential Functions1 min

1.43.2. Example2 min

1.43.3. Practice Level 33 min

1.44.1. Basics of Logarithms3 min

1.44.2. Example3 min

1.44.3. Practice Level 1

1.44.4. Practice Level 2

1.44.5. Practice Level 3

1.45.1. Logarithm Laws4 min

1.45.2. Example2 min

1.45.3. Example 2 min

1.45.4. Practice Level 1

1.45.5. Practice Level 2

1.45.6. Practice Level 32 min

1.46.1. Solving Logarithmic Equations2 min

1.46.2. Example2 min

1.46.3. Practice Level 1

1.46.4. Practice Level 22 min

1.46.5. Practice Level 31 min

1.47.1. Using Logarithms to Solve Exponential Equations1 min

1.47.2. Example58 sec

1.47.3. Practice Level 1

1.47.4. Practice Level 2

1.47.5. Practice Level 3

1.48.1. Characteristics of Logarithmic Functions2 min

1.48.2. Example57 sec

1.48.3. Practice Level 1

1.48.4. Practice Level 2

1.48.5. Practice Level 3

1.49.1. Transformations of Logarithmic Functions2 min

1.49.2. Example2 min

1.49.3. Practice Level 1

1.49.4. Practice Level 2

1.49.5. Practice Level 3

1.50.1. Applications of Logarithmic Functions30 sec

1.50.2. Example

1.50.3. Practice Level 2

2. Systems of Linear Equations

2.3hr

2.1.1. Systems of Linear Equations4 min

2.1.2. Augmented Matrix4 min

2.1.3. Example: Augmented Matrix5 min

2.1.4. Practice: Augmented Matrix4 min

2.1.5. Practice: Augmented Matrix3 min

2.2.1. Performing Elementary Row Operations8 min

2.2.2. Example: Performing Elementary Row Operations5 min

2.2.3. Practice: Performing Elementary Row Operations3 min

2.2.4. Practice: Performing Elementary Row Operations6 min

2.2.5. Practice: Performing Elementary Row Operations7 min

2.3.1. Row Echelon Form4 min

2.3.2. Example: Row Echelon Form5 min

2.3.3. Practice: Row Echelon Form5 min

2.3.4. Example: Interpreting Row Echelon Form16 min

2.3.5. Practice: Interpreting Row Echelon Form

2.3.6. Reduced Row Echelon Form4 min

2.3.7. Example: Reduced Row Echelon Form5 min

2.3.8. Practice: Reduced Row Echelon Form5 min

2.4.1. Transforming an Augmented Matrix to Reduced Row Echelon Form4 min

2.4.2. Example: Transforming an Augmented Matrix to RREF14 min

2.4.3. Practice: Transforming an Augmented Matrix to RREF14 min

2.4.4. Solutions From RREF9 min

2.4.5. Example: Solutions From RREF5 min

2.4.6. Practice: Solutions From RREF

2.Q. Systems of Linear Equations Checkpoint Quiz2 Qns

3. Matrix Operations

1.7hr

3.1.1. Matrix Basics 7 min

3.1.2. Matrix Addition (and Subtraction)3 min

3.1.3. Scalar Multiplication5 min

3.2.1. Matrix Multiplication14 min

3.2.2. Matrix Powers3 min

3.2.3. Example: Matrix Multiplication8 min

3.2.4. Practice: Matrix Multiplication

3.2.5. Practice: Matrix Multiplication

3.2.6. Practice: Matrix Multiplication

3.2.7. Matrix-Vector Form of a Linear System8 min

3.3.1. Matrix Transpose8 min

3.3.2. Example: Matrix Transpose3 min

3.3.3. Symmetric Matrices6 min

3.3.4. Example: Symmetric Matrices4 min

3.3.5. Practice: Symmetric Matrices

3.4.1. Matrix Inverse11 min

3.4.2. Example: Matrix Inverse4 min

3.4.3. Practice: Matrix Inverse

3.4.4. Example: Solving a Linear System with the Inverse7 min

3.5.1. Matrix Inverse Algorithm5 min

3.5.2. Example: Matrix Inverse Algorithm7 min

3.5.3. Practice: Matrix Inverse Algorithm

4. Limits

1.6hr

4.1.1. Limit Basics2 min

4.1.2. Example1 min

4.2.1. Limit Laws 2 min

4.2.2. Practice1 min

4.3.1. One-Sided Limits2 min

4.3.2. Example: One-Sided Limit2 min

4.3.3. Practice: One-Sided Limits3 min

4.4.1. Infinite Limits2 min

4.4.2. Example1 min

4.4.3. Example3 min

4.5.1. Limits at Infinity3 min

4.5.2. Example60 sec

4.5.3. Example3 min

4.5.4. Example4 min

4.5.5. Practice 3 min

4.5.6. Practice3 min

4.6.1. Indeterminate Forms50 sec

4.7.1. Computing Limits by Factoring1 min

4.7.2. Example1 min

4.7.3. Practice2 min

4.8.1. Computing Limits by Multiplying by the Conjugate3 min

4.8.2. Example3 min

4.8.3. Practice3 min

4.9.1. Computing Limits by Getting a Common Denominator1 min

4.9.2. Example2 min

4.9.3. Practice4 min

4.10.1. The Squeeze Theorem2 min

4.10.2. Example2 min

4.10.3. Practice2 min

4.11.1. Continuity2 min

4.11.2. Example2 min

4.11.3. Practice4 min

5. Derivatives

2.1hr

5.1.1. Derivatives as a Rate of Change2 min

5.1.2. Average Rate of Change46 sec

5.1.3. Example2 min

5.1.4. Practice1 min

5.2.1. The Limit Definition of a Derivative2 min

5.2.2. Example3 min

5.2.3. Practice5 min

5.2.4. When is Function Not Differentiable?

5.3.1. Differentiation Laws51 sec

5.4.1. The Power Rule1 min

5.4.2. Example2 min

5.4.3. Practice51 sec

5.5.1. Tangent Lines1 min

5.5.2. Example3 min

5.5.3. Practice2 min

5.6.1. Normal Lines1 min

5.6.2. Example3 min

5.6.3. Practice1 min

5.7.1. The Product Rule49 sec

5.7.2. Example2 min

5.7.3. Example2 min

5.7.4. Practice2 min

5.8.1. The Quotient Rule1 min

5.8.2. Example2 min

5.8.3. Practice4 min

5.9.1. The Chain Rule1 min

5.9.2. Example2 min

5.9.3. Practice3 min

5.10.1. Derivatives of Exponential Functions1 min

5.10.2. Example52 sec

5.10.3. Example3 min

5.10.4. Practice1 min

5.11.1. Higher Order Derivatives2 min

5.11.2. Example3 min

5.11.3. Practice2 min

5.11.4. Practice2 min

5.12.1. Derivatives of Logarithmic Functions1 min

5.12.2. Example56 sec

5.12.3. Example2 min

5.12.4. Practice2 min

5.12.5. Practice: Log and Ln2 min

5.13.1. Logarithmic Differentiation2 min

5.13.2. Example2 min

5.13.3. Practice3 min

6. Applications of Differentiation

3hr

6.1.1. Differentials2 min

6.1.2. Example3 min

6.1.3. Practice2 min

6.1.4. Linearization (Linear Approximation)59 sec

6.1.5. Example3 min

6.1.6. Practice3 min

6.1.7. Practice3 min

6.2.1. Intervals of Increase and Decrease1 min

6.2.2. Example2 min

6.2.3. Practice3 min

6.3.1. Critical Points and Extrema2 min

6.3.3. Practice3 min

6.4.1. 1st and 2nd Derivative Tests3 min

6.4.2. Example7 min

6.4.4. Practice3 min

6.5.1. Maximum and Minimum on Closed Intervals2 min

6.5.2. Example4 min

6.5.3. Practice3 min

6.5.4. Practice3 min

6.6.1. Concavity and Inflection Points56 sec

6.6.2. Example6 min

6.6.3. Practice6 min

6.7.1. Curve Sketching2 min

6.7.2. Example7 min

6.7.3. Practice9 min

6.7.4. Practice

6.8.1. Optimization1 min

6.8.2. Example4 min

6.8.3. Example4 min

6.8.4. Practice6 min

6.8.5. Practice4 min

6.9.1. Related Rates

6.9.2. Related Rates5 min

6.9.3. Related Rates5 min

6.9.4. Linear Approximation4 min

6.9.5. Linear Approximation

6.9.6. Taylor Series from Definition7 min

6.9.7. Taylor Polynomials

6.9.8. Maclaurin Polynomial

6.9.9. Newton's Method2 min

6.9.10. Newton's Method

6.9.11. L'Hopital's Rule

6.9.12. L'Hopital's Rule

6.9.13. L'Hopital's Rule

6.9.15. L'Hopital's Rule

6.9.16. L'Hopital's Rule

6.9.19. Extreme Value Theorem

6.9.20. Rolle's Theorem1 min

6.9.21. Rolle's Theorem6 min

6.9.23. MVT2 min

6.9.24. MVT3 min

6.9.26. Intervals of Increase and Decrease

6.9.27. Intervals of Increase and Decrease3 min

6.9.28. Critical Points6 min

6.9.29. Critical Points1 min

6.9.30. Extrema5 min

6.9.31. Extrema6 min

6.9.32. Second Derivative Test10 min

6.9.33. Curve Sketching

6.9.34. Curve Sketching12 min

6.9.35. Curve Sketching

6.9.36. Optimization

6.9.37. Optimization

6.9.38. Optimization

7. Applications of Differentiation for Business & Econ

42min

7.1.1. Cost, Revenue, and Profit2 min

7.1.2. Example3 min

7.1.3. Practice2 min

7.1.4. Practice4 min

7.2.1. Elasticity of Demand2 min

7.2.2. Example3 min

7.2.3. Practice: Elasticity of Demand4 min

7.3.1. Compound Interest2 min

7.3.2. Example3 min

7.3.3. Practice2 min

7.4.1. Elasticity of Demand

7.4.2. Elasticity of Demand

7.4.3. Profit5 min

7.4.4. Optimization w/ Profit6 min

7.4.7. Compound Interest3 min

7.4.8. Compound Interest

8. Integrals

2.3hr

8.1.1. Antiderivatives (Indefinite Integral)3 min

8.1.2. Example2 min

8.1.3. Practice1 min

8.1.4. Practice1 min

8.1.5. Practice1 min

8.2.1. Integration by Substitution2 min

8.2.2. Example4 min

8.2.3. Example3 min

8.2.4. Example (w/ Back-Substitution)4 min

8.2.5. Practice

8.2.6. Practice

8.2.7. Practice

8.2.8. Practice (w/ Back-Substitution)

8.3.1. Antiderivatives (Indefinite Integral) of Exponential Functions1 min

8.3.2. Example42 sec

8.3.3. Practice1 min

8.3.4. Practice2 min

8.4.1. Antiderivatives (Indefinite Integral) of

\frac{1}{x}

8.4.2. Example2 min

8.4.3. Practice1 min

8.5.1. Finite Sums & Summation Notation3 min

8.5.2. Example5 min

8.5.3. Practice4 min

8.5.4. Practice4 min

8.6.1. Approximating Areas & Riemann Sums 3 min

8.6.2. Example4 min

8.6.3. Practice3 min

8.7.1. Fundamental Theorem of Calculus - Part I2 min

8.7.2. Example1 min

8.7.3. Example48 sec

8.7.4. Practice1 min

8.7.5. Practice2 min

8.8.1. Fundamental Theorem of Calculus - Part 21 min

8.8.2. Example2 min

8.8.4. Practice 4 min

8.8.5. Practice3 min

8.8.6. Practice1 min

8.9.1. The Definite Integrals & Area Under a Curve2 min

8.9.2. Example7 min

8.9.3. Practice2 min

8.9.4. Practice12 min

8.10.1. Properties of the Definite Integral2 min

8.10.2. Practice2 min

8.10.3. Practice2 min

8.11.1. Computing Definite Integrals Geometrically44 sec

8.11.2. Example1 min

8.11.3. Practice1 min

9. Applications of Integration

1.2hr

9.I. Chapter Intro23 sec

9.1.1. Area between curves1 min

9.1.2. Example5 min

9.1.3. Example5 min

9.1.4. Practice

10. Partial Derivatives

1.7hr

10.1.1. Basics of Dot Product1 min

10.1.2. Example: Dot product1 min

10.1.3. Dot product properties3 min

10.1.4. Example: Orthogonal Vectors1 min

10.1.5. Practice Question: Properties of Dot Product2 min

10.1.6. Practice Question: Orthogonal Vectors53 sec

10.1.7. Dot product and angles1 min

10.1.8. Example: Dot product and Angles1 min

10.1.9. Practice Question: Dot Product and angle1 min

10.1.10. Planes in R32 min

10.1.11. Example: Planes in R32 min

10.1.12. Practice Question: Planes in R32 min

10.1.13. Point of intersection between a line and a plane2 min

10.1.14. Example: Intersection Between a Line and a Plane2 min

10.1.15. Practice Question: Intersection of Lines and Planes57 sec

10.1.16. Example: Thales' Theorem5 min

10.2.1. Partial Derivatives3 min

10.2.2. Higher Order Partial Derivatives & Differentials2 min

10.2.3. Example1 min

10.2.4. Example1 min

10.2.5. Practice1 min

10.2.6. Practice2 min

10.3.1. Chain Rule2 min

10.3.2. Chain Rule & Implicit Differentiation3 min

10.3.3. Example2 min

10.3.4. Example1 min

10.3.5. Practice2 min

10.3.6. Practice

10.4.1. Maximums, Minimums, and Critical Points5 min

10.4.2. Example5 min

10.4.3. Example5 min

10.4.4. Practice4 min

10.4.5. Practice3 min

10.5.1. Absolute Max and Min2 min

10.5.2. Example3 min

10.5.3. Example6 min

10.5.4. Practice6 min

10.6.1. Lagrange Multipliers3 min

10.6.2. Example3 min

10.6.3. Example

10.6.4. Practice3 min

10.6.5. Practice4 min

11. Differential Equations

1.2hr

11.1.1. Differential Equations8 min

11.1.2. Direction Fields/Slope Fields5 min

11.1.3. Practice Question: DE Solution Graphs3 min

11.2.1. Separable First Order Differential Equations3 min

11.2.2. Practice (separable DE)2 min

11.2.3. Practice (IVP)3 min

11.2.4. Practice (IVP)3 min

11.2.5. Practice (IVP*)5 min

11.2.6. Practice (separable DE)1 min

11.3.1. Linear First Order Differential Equations7 min

11.3.2. Practice (integrating factor)1 min

11.3.3. Practice (linear DE)3 min

11.3.4. Practice (IVP)3 min

11.3.5. Practice (IVP)4 min

11.4.1. Exponential Growth and Decay3 min

11.4.2. Example4 min

11.4.3. Example5 min

11.4.4. Practice5 min

11.4.5. Practice (population)3 min

11.4.6. Practice (population)4 min

11.5.1. Second-Order Reducible DE

11.5.2. Practice: Dependent Variable Missing

11.5.3. Practice: Independent Variable Missing

11.Q. Exam-Like Practice Questions4 Qns

I Welcome

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Welcome to Calculus!
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Answered

Anonymous

What are the conecpts to focus on for midterm 1 at western 

C

Corey M

Instructor

Likely limits and differentiation techniques through the chain rule, but double-check with your instructor to see exactly what will be covered.

Unanswered

L

Layan E

Use the Intermediate Value Theorem to show that thereis a root of the given equation in the specified interval.Use the Intermediate Value Theorem to show that thereis a root of the given equation in the specified interval.
sin(x)=x2−x, x∈(1,2)

C

Corey M

Instructor

While this isn't quite the place for this question (please refer to the IVT section in the course), and we can't really just solve random problems for you, I can give you a bit of a hint:  You could try moving everything to one side of the equation and treating it like a function, and then see if you can't find function values within your specified range that return a positive value and a negative value (another hint: try the endpoints of your interval first).  If you're able to do that, then the IVT tells us that there should exist a function input between those two points that returns 0 or, in other words, that is a root.

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