Course Syllabus

for

 “Introduction to Calculus and Analysis I”

 

Course Description:

Calculus is a foundational course at Tongji School of Pharmcy, HUST. It plays an important role in the understanding of core theories in pharmaceutical sciences, particularly for physical chemistry, analytical chemistry, pharmacokinetics, computational drug design and development, etc. This introductory calculus course covers integration anddifferentiation of functions of one variable, with both analytical and numerical solutions. Topics include:

  1. Concepts of Number, Function, Sequence, Limits and Continuity
  2. Definite and Indefinite Integration
  3. Differentiation Rules, Application to Extremum Problems
  4. The Fundamental Theorem of Calculus
  5. Numerical Methods for Computing Integrals

 

Prerequisites:

The prerequisites are high school algebra and trigonometry. Prior experience with calculus is helpful but not essential.

 

Textbook:

Richard Courant, Fritz John, Introduction to Calculus and Analysis I (Reprinted in 1989), Springer-Verlag New York, Inc.

 

Course Objectives:

After completing this course, students should get the idea of connection between Mathematic concepts/theories and reality applications, and also should have developed a clear understanding of the fundamental concepts of single variable calculus and a range of skills allowing them to work effectively to:

  1. Evaluate the limits of functions.
  2. State whether a function is continuous or discontinuous based on both the graph and the definition of continuity.
  3. Integrate rational, algebraic, trigonometric, logarithm, exponential, and composite functions.
  4. Differentiate rational, algebraic, trigonometric, logarithm, exponential, and composite functions.
  5. Compute definite integrals by the Fundamental Theorem of Calculus, by substitution, and by numerical techniques.

 

Content Outline:

  1. Basic Concepts in Mathematics:
    1. The continuum of numbers
      1. Natural, Fractional, Rational, Irrational, Real numbers
      2. Graphical representation of numbers
      3. Neighborhood and nested intervals
    2. The concept of functions
      1. Domain and Range of a function
      2. Mapping-Graph
      3. Graphical representation of function
      4. Continuity
    3. The elementary functions
      1. Rational, Algebraic, Trigonometric, Logarithm, and Exponential functions
      2. Compound functions
      3. Sequences
  2. Using Limits
    1. Mathematical Induction
    2. Evaluation of limits
      1. Rational operations with limits
      2. Evaluate the limit of a function at a point both algebraically and graphically
      3. Evaluate the limit of a function at infinity both algebraically and graphically.
      4. Use the limit to determine the continuity of a function.
  3. Finding Integrals
    1. Integral Definition
      1. The number π as a limit
      2. Approximation of an area
      3. The integral as an area
      4. Analytical definition of integral and notations
    2. Integral Evaluation
      1. Table of elementary integrals
      2. Fundamental rules of integration
      3. Integrate algebraic, natural exponential, natural logarithm, trigonometric
      4. Definite and indefinite integration
  4. Finding Derivatives
    1. Derivative Definition
      1. Tangent
      2. Velocity
    2. The Fundamental Theorems of the Calculus
    3. Derivative Evaluation
      1. Rules for differentiation
      2. The chain rule for differentiation of composite functions
      3. Find derivatives involving powers, exponents, trigonometric, exponential and logarithmic functions
      4. Find the maxima and minima of functions
  5. Advanced Techniques of Integration
    1. Method of Substitution
    2. Integration by Pats
    3. Numerical Computation of Integrals
      1. Approximation by rectangles
      2. Refined approximation: Trapezoidal Rule, Simpson’s Rule, Gaussian Quadrature.

 

Grading:

Homework will be assigned on most days. Quizzes will be administered at the conclusion of each part. A final examination covers the entire course. The score for this course consists of marks from homework (25%), quiz (10%), attendance (15%), and final examination (50%).