Description:
Basic concepts of calculus; limits, continuity, and differentiability of functions; differentiation of algebraic and transcendental functions involving one or more variables; applications of differential calculus to problems on optimization, rates of change, related rates, tangents and normals, and approximations; partial differentiation and transcendental curve tracing.
Overview:
Calculus is a branch of mathematics that involves the study of rates of change. Before calculus was invented, all math was static: It could only help calculate objects that were perfectly still. But the universe is constantly moving and changing. No objects—from the stars in space to subatomic particles or cells in the body—are always at rest. Indeed, just about everything in the universe is constantly moving. Calculus helped to determine how particles, stars, and matter actually move and change in real-time.
Calculus is used in a multitude of fields that you wouldn't ordinarily think would make use of its concepts. Among them are physics, engineering, economics, statistics, and medicine. Calculus is also used in such disparate areas as space travel, as well as determining how medications interact with the body, and even how to build safer structures. You'll understand why calculus is useful in so many areas if you know a bit about its history as well as what it is designed to do and measure (ThoughtCo., 2020).
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(1) What Is Calculus? Definition and Practical Applications
Watch:
(1) What is Calculus? (Mathematics)
a. Definitions
b. Classification of Functions
c. Domain and Range of a Function
d. Graph of a Function
e. Functional Notation
f. Evaluation of a Function
g. Combinations of Functions
h. One-Valued and Many-Valued Functions
i. Odd and Even Functions
j. Special Function Types
k. Functions as Mathematical Models
2. Continuity and Limits
a. Definition and Properties of Continuous Functions
b. Notion of a Limit
c. Properties of Limits
d. Operations with Limits
e. Evaluation of Limits
f. One-Sided Limits
g. Unbounded Functions
3. The Derivative
a. Definition and Notion of the Derivative
b. Determination of the Derivative by Increments
c. Differentiation Rules
4. The Slope
a. Definition of Slope as the Derivative of a Function
b. Determination of the Slope of a Curve at a Given Point
5. Rate of Change
a. Average Rate of Change
b. Instantaneous Rate of Change
6. The Chain Rule and the General Power Rule
7. Implicit Differentiation
8. Higher-Order Derivatives
9. Polynomial Curves
a. Generalities About Straight Lines
b. Tangents and Normal to Curves
c. Extrema and the First Derivative Test
d. Concavity and the Second Derivative Test
e. Points of Inflection
f. Sketching Polynomial Curves
10. Applications of the Derivative
a. Optimization Problems
b. Related Rates
11. The Differential
a. Definition and Applications of the Differential
b. Error Propagation
c. Approximate Formulas
12. Derivatives of Trigonometric Functions
a. Definition and Elementary Properties
b. Graphs of Trigonometric Functions
c. Applications
13. Derivative of Inverse Trigonometric Functions
a. Definition and Elementary Properties
b. Graphs of Inverse Trigonometric Functions
c. Applications
14. Derivative of Logarithmic and Exponential Functions
a. Definition and Elementary Properties
b. Graphs of Logarithmic and Exponential Functions
c. Applications
15. Derivative of the Hyperbolic Functions
a. Definition and Elementary Properties
b. Graphs of Hyperbolic Functions
c. Applications
16. Solutions of Equations
a. Newton's Method of Approximation
b. Newton-Raphson Law
17. Transcendental Curve Tracing
a. Logarithmic and Exponential Functions
18. Parametric Equations
19. Partial Differentiation
(1) What Is Calculus? Definition and Practical Applications
Watch:
(1) What is Calculus? (Mathematics)
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Topics:
1. Functionsa. Definitions
b. Classification of Functions
c. Domain and Range of a Function
d. Graph of a Function
e. Functional Notation
f. Evaluation of a Function
g. Combinations of Functions
h. One-Valued and Many-Valued Functions
i. Odd and Even Functions
j. Special Function Types
k. Functions as Mathematical Models
2. Continuity and Limits
a. Definition and Properties of Continuous Functions
b. Notion of a Limit
c. Properties of Limits
d. Operations with Limits
e. Evaluation of Limits
f. One-Sided Limits
g. Unbounded Functions
3. The Derivative
a. Definition and Notion of the Derivative
b. Determination of the Derivative by Increments
c. Differentiation Rules
4. The Slope
a. Definition of Slope as the Derivative of a Function
b. Determination of the Slope of a Curve at a Given Point
5. Rate of Change
a. Average Rate of Change
b. Instantaneous Rate of Change
6. The Chain Rule and the General Power Rule
7. Implicit Differentiation
8. Higher-Order Derivatives
9. Polynomial Curves
a. Generalities About Straight Lines
b. Tangents and Normal to Curves
c. Extrema and the First Derivative Test
d. Concavity and the Second Derivative Test
e. Points of Inflection
f. Sketching Polynomial Curves
10. Applications of the Derivative
a. Optimization Problems
b. Related Rates
11. The Differential
a. Definition and Applications of the Differential
b. Error Propagation
c. Approximate Formulas
12. Derivatives of Trigonometric Functions
a. Definition and Elementary Properties
b. Graphs of Trigonometric Functions
c. Applications
13. Derivative of Inverse Trigonometric Functions
a. Definition and Elementary Properties
b. Graphs of Inverse Trigonometric Functions
c. Applications
14. Derivative of Logarithmic and Exponential Functions
a. Definition and Elementary Properties
b. Graphs of Logarithmic and Exponential Functions
c. Applications
15. Derivative of the Hyperbolic Functions
a. Definition and Elementary Properties
b. Graphs of Hyperbolic Functions
c. Applications
16. Solutions of Equations
a. Newton's Method of Approximation
b. Newton-Raphson Law
17. Transcendental Curve Tracing
a. Logarithmic and Exponential Functions
18. Parametric Equations
19. Partial Differentiation
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