Limits, Continuity and Differentiability of Real-Valued Functions - Advanced Calculus

This course comprehensively treats limits, continuity and differentiability of real-valued single-variable functions. You will learn how to: - Define and classify real-valued functions and their properties, such as domain, range, and graphs - Identify and compare different kinds of real-valued functions, such as composites, piecewise functions, polynomials, rational functions, algebraic functions, transcendental functions, odd and even functions - Find the limit of a function as the independent variable approaches a certain value, and use the formal (epsilon-delta) and informal definitions of limits - Find the limit of a function using various methods and techniques, such as direct substitution, theorems, graphing, factorization, conjugates, L’Hôpital’s rule, use of known special limits, the squeeze theorem, two-sided limits, etc. - Find the limit of a function at infinity or an infinite limit, and use the formal and informal definitions of limits at infinity and infinite limits - Find the continuity of a function at a point or on an interval, and use the graphical, formal, and informal definitions of continuity - Identify and understand different types of discontinuities, such as removable, jump, and infinite discontinuities - Apply the max-min theorem and the intermediate-value theorem to continuous functions and their graphs - Find the differentiability of a function at a point or on an interval, and use the graphical and formal definitions of differentiability - Find the derivative of a function as the slope of the tangent line or the rate of change of the function, and use the first-principle method and the rules of differentiation to calculate derivatives - Apply the mean-value theorem and the Rolle's theorem to differentiable functions and their graphs - Find the higher-order derivatives of a function by applying the differentiation rules repeatedly, and use the notation and terminology for higher derivatives - Apply the Leibniz's formula to find the derivatives of products of functions - Find the Taylor and Maclaurin series expansions of differentiable functions and use them to approximate functions and their values This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of the course, you will have a firm grasp of advanced single-variable differential calculus concepts and their applications. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve analyzing the behavior, trends, and optimization of functions and their graphs. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

139

50 hrs

$ 10.00

Payment required for enrolment
Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
MTH 201: Mathematical Methods I
MTH 201: Mathematical Methods I
Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

1
Real Numbers

What are real numbers, and how do they differ from natural numbers, integers, rational numbers, irrational numbers? Meaning of infinity; defining finite and infinite intervals.

Chapter lessons

1.Number systems18:36

An explanation of natural numbers, integers, rational numbers, irrational numbers, real numbers and complex numbers.

2.Rationals and irrationals26:08

A closer look at differences between rational and irrational numbers.

3.Intervals16:55

Meaning and examples of intervals on the real line.

4.Infinity14:44

Meaning and use of infinity.

2
Real-Valued Functions

Real-valued functions - meaning, domain, range and graphs.

Chapter lessons

1.Introduction24:36

Meaning of functions and real-valued functions.

2.Domain of functions31:32

Domain of functions and its calculation.

3.Range of functions6:17

Meaning of the range of functions.

4.Worked examples (1)24:55

More worked examples on the domain of real-valued functions.

3
Kinds of Real-Valued Functions

A look at different kinds of real-valued functions - composites, piecewise functions, polynomials, rational functions, algebraic functions, transcendental functions, odd and even functions.

Chapter lessons

1.Polynomials12:01

Meaning, domain and examples of polynomials.

2.Rational functions7:47

Meaning, domain and examples of rational functions.

3.Algebraic functions5:19

Meaning, domain and examples of algebraic functions.

4.Piecewise-defined functions16:00

Meaning, domain and examples of piecewise-defined functions.

5.Transcendental functions5:30

An introduction to transcendental functions.

6.Exponential and logarithmic functions17:17

Meaning, domain and examples of exponential and logarithmic functions.

7.Trigonometric functions21:24

Meaning, domain and examples of trigonometric functions.

8.Inverse trigonometric functions7:37

Meaning, domain and examples of inverse trigonometric functions.

9.Hyperbolic functions20:21

Meaning, domain and examples of hyperbolic functions.

10.Inverse hyperbolic functions19:37

Meaning, domain and examples of inverse hyperbolic functions.

11.Odd and even functions4:05

Meaning and examples of odd and even functions.

4
Limits of Functions (Definition)

Meaning of the limits of a real-valued functions - graphical illustrations and formal (epsilon-delta) definitions.

Chapter lessons

1.Informal definition19:20

An informal definition of the limit of real-valued functions.

2.Formal definition17:26

A formal (rigorous) definition of the limit of real-valued functions.

3.Worked examples I13:51

Worked examples on the proof of limits of functions.

4.Worked examples II41:12

Worked examples on the proof of limits of functions.

5.Worked examples III18:09

Worked examples on the proof of limits of functions.

6.Worked examples IV23:23

Worked examples on the proof of limits of functions.

7.Worked examples V10:26

Worked examples on the proof of limits of functions.

5
Limits of Functions (Evaluation)

Different methods of evaluating limits of real-valued functions - direct substitution, theorems, graphing, factorization, conjugates, L’Hôpital’s rule, use of known special limits, squeeze theorem, limits of piecewise-defined functions, etc.

Chapter lessons

1.Direct substitution23:58

Evaluating limits of functions by direct substitution.

2.Theorems32:03

Evaluating limits of functions by use of fundamental theorems.

3.Graphing35:01

Evaluating limits of functions by graphing.

4.Factorization19:30

Evaluating limits of functions by factorization.

5.Conjugates13:00

Evaluating limits of functions by use of conjugates.

6.L’Hôpital’s rule45:15

Evaluating limits of functions by L’Hôpital’s rule.

7.Special limits27:58

Evaluating limits of functions by use of known special limits.

8.Worked examples I20:28

Worked examples on evaluation of limits.

9.Worked examples II27:56

Worked examples on evaluation of limits.

10.The squeeze theorem34:06

Evaluating limits using the squeeze theorem.

11.Piecewise-defined functions22:20

Evaluating limits of piecewise-defined functions.

12.What if everything fails?45:39

Examining limits for which the L'Hospital's rule fails.

13.Worked examples III19:34

More worked examples on evaluation of limits.

14.Worked examples IV35:49

More worked examples on evaluation of limits.

6
Limits of Functions (Infinity)

Formal and informal definitions of limits at infinity and infinite limits; different methods of evaluating limits of real-valued functions at infinity and infinite limits.

Chapter lessons

1.Informal definition42:51

Informal definition of limits at infinity and infinite limits.

2.Formal definition15:13

Formal (rigorous) definition of limits at infinity and infinite limits.

3.Worked examples I40:53

Worked examples on formal definition of limits at infinity.

4.Worked examples II22:49

Worked examples on formal definition of infinite limits.

5.Evaluation of limits at infinity I30:49

Evaluation of limits at infinity for polynomials and rational functions.

6.Evaluation of limits at infinity II30:11

Evaluation of limits of exponential functions at infinity.

7.Evaluation of limits at infinity III20:30

Evaluation of limits at infinity using a special limit.

8.Evaluation of limits at infinity IV21:03

Evaluation of the limits of functions with rationalizable radicals at infinity.

9.Evaluation of limits at infinity V30:19

Evaluation of the limits of functions with non-rationalizable radicals at infinity.

10.Indeterminate forms I12:41

Indeterminate form ∞-∞.

11.Indeterminate forms II38:19

Indeterminate form 0*∞.

12.Indeterminate forms III33:42

Indeterminate forms involving powers.

13.Worked examples III13:38

Worked examples on indeterminate forms.

14.Worked examples IV18:38

More worked examples on indeterminate forms.

15.Worked examples V15:12

More worked examples on indeterminate forms.

7
Continuity of Functions (Definition)

Continuity (at an interior point, at an endpoint, on an interval) of real-valued functions - graphical illustration, formal and informal definitions.

Chapter lessons

1.Informal definition16:07

Informal definition of continuity.

2.Continuity at an interior point13:34

Continuity at an interior point of the domain of a function.

3.Continuity at an endpoint8:53

Continuity at an endpoint of the domain of a function.

4.Continuity on an interval17:22

Continuity of a function on an interval in its domain.

5.Examples of continuous functions24:12

Some examples of functions that are continuous everywhere in their domain of definition.

6.Worked examples I34:37

Worked examples on continuity of functions.

7.Worked examples II15:25

Worked examples on continuity of functions.

8.Worked examples III18:33

Worked examples on continuity of functions.

9.Worked examples IV23:03

Worked examples on continuity of functions.

8
Continuity of Functions (Types, Theorems)

Understanding removable discontinuities, the max-min theorem and the intermediate-value theorem.

Chapter lessons

1.Removable discontinuities10:52

Meaning of removable and non-removable discontinuities.

2.Worked examples I13:12

Worked examples on removable discontinuities.

3.The max-min theorem11:12

Understanding the max-min theorem.

4.The intermediate-value theorem11:20

Understanding the intermediate-value theorem.

5.Worked examples II12:44

Worked examples on the intermediate-value theorem.

9
Differentiability of Functions (Definition)

Meaning of the differentiability of real-valued functions at a point; illustration by slope of a straight line.

Chapter lessons

1.Slope of a line8:58

A review of the meaning of the slope (gradient) of a straight line.

2.Differentiability at a point11:08

Differentiability of a function, and its derivative at a given point.

3.Worked examples I10:19

Some examples on the determination of the derivative of a function at a given point.

4.Worked examples II17:37

More examples on the determination of the derivative of a function at a given point.

10
Differentiability of Functions (Derivatives)

Differentiability on an interval; relation of differentiability and continuity; review of how to find the derivatives of real-valued functions; rules of derivatives of sums, products and quotients of functions and their proofs.

Chapter lessons

1.Differentiability on an interval17:15

Derivative of a function over an interval.

2.Differentiability and its relation to continuity18:39

How continuity and differentiability are related.

3.Worked examples I53:13

Worked examples on the derivative of a function over an interval.

4.Worked examples II18:36

More worked examples on the derivative of a function over an interval.

5.Rules of differentiation I24:52

Rules of differentiation of functions.

6.Rules of differentiation II25:58

Rules of differentiation of functions.

7.Rules of differentiation III35:43

Rules of differentiation of functions.

8.Worked examples III23:07

Worked examples on differentiation of functions.

11
Differentiability of Functions (Theorems)

Understanding the mean-value and Rolle's theorems.

Chapter lessons

1.Rolle's theorem57:35

The Rolle's theorem and its implications.

2.The mean-value theorem35:28

The mean-value theorem and its implications

3.Worked examples I28:29

Worked examples on the Rolle's and mean-value theorems.

4.Worked examples II29:53

Worked examples on the Rolle's and mean-value theorems.

5.Worked examples III25:43

Worked examples on the Rolle's and mean-value theorems.

12
Differentiability of Functions (Higher-Order Derivatives)

Higher-order derivatives of differentiable functions - meaning, proof of Leibniz's formula and its applications.

Chapter lessons

1.Definition44:24

Meaning of higher-order derivatives and how to evaluate them.

2.Worked examples I22:07

Worked examples on the evaluation of higher-order derivatives.

3.Worked examples II1:07:31

Worked examples on evaluation of higher-order derivatives.

4.Leibniz's formula1:18:57

Evaluating higher-order derivatives of a product of functions.

5.Worked examples III36:54

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

6.Worked examples IV50:11

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

7.Worked examples V27:31

Worked examples on evaluation of higher-order derivatives using the Leibnitz's formula.

13
Taylor and Maclaurin Series

Taylor and Maclaurin series expansion of differentiable functions.

Chapter lessons

1.Maclaurin polynomials1:26:27

Polynomial approximations of differentiable functions.

2.Taylor polynomials35:41

Polynomial approximations of differentiable functions.

3.Taylor and Maclaurin series7:38

Infinite series representation of differentiable functions.

4.Worked examples I26:11

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.

5.Worked examples II41:44

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.

6.Worked examples III41:18

Worked examples on Taylor and Maclaurin series expansion of differentiable functions.