Mathematics I Calculus Syllabus Complete Details & Topics
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Course Overview: The course covers the concepts of functions, limits, continuity, differentiation, integration of functions of one variable, logarithmic and exponential functions, applications of derivatives and antiderivatives, differential equations, vectors and their applications, partial derivatives, and multiple integrals.
Course Objectives
The objective of this course is to make students able to:
- Understand and formulate real-world problems into mathematical statements.
- Develop solutions to mathematical problems at the level appropriate to the course.
- Describe or demonstrate mathematical solutions either numerically or graphically
Course Contents
Unit 1: Function of One Variable (5 Hrs.)
- Four ways of representing a function
- Linear mathematical model
- Polynomial, Rational, Trigonometric, Exponential, and Logarithmic functions
- Combination of functions
- Range and domain of functions and their graphs
Unit 2: Limits and Continuity (4 Hrs.)
- Precise definition of limit
- Limits at infinity
- Continuity
- Horizontal asymptotes
- Vertical and slant asymptotes
Unit 3: Derivatives (4 Hrs.)
- Tangents and velocity
- Rate of change
- Review of derivative
- Differentiability of a function
- Mean value theorem
- Indeterminate forms and L’Hospital’s rule
Unit 4: Applications of Derivatives (4 Hrs.)
- Curve sketching
- Review of maxima and minima of one variable
- Optimization problems
- Newton’s method
Unit 5: Antiderivatives (5 Hrs.)
- Review of antiderivatives
- Rectilinear motion
- Indefinite integrals and net change
- Definite integral
- The Fundamental Theorem of Calculus
- Improper integrals
Unit 6: Applications of Antiderivatives (5 Hrs.)
- Areas between the curves
- Volumes of cylindrical cells
- Approximate integrations
- Arc length
- Area of the surface of revolution
Unit 7: Ordinary Differential Equations (6 Hrs.)
- Introduction to first-order equations
- Separable equations
- Linear equations
- Second-order linear differential equations
- Non-homogeneous linear equations
- Method of undetermined coefficients
Unit 8: Infinite Sequence and Series (5 Hrs.)
- Infinite sequence and series
- Convergence tests and power series
- Taylor’s and Maclaurin’s series
Unit 9: Plane and Space Vectors (4 Hrs.)
- Introduction to vectors
- Applications of vectors
- Dot product and cross-product
- Equations of lines and planes
- Derivative and integrals of vector functions
- Arc length and curvature
- Normal and binormal vectors
- Motion in space
Unit 10: Partial Derivatives and Multiple Integrals (3 Hrs.)
- Limit and continuity
- Partial derivatives
- Tangent planes
- Maximum and minimum values
- Multiple integrals
Recommended Books
Textbook:
- Calculus Early Transcendentals, James Stewart, 7E, CENGAGE Learning.
Reference Books:
- Calculus Early Transcendentals, Thomas, 12th Edition, Addison Wesley