Topic outline

  • General

    Calculus I

    Spring 2013/2014


     Topics for the Midterm Examination

    1. Techniques for computing limits.

    2. Infinite llimits.

    3. Limits at infinity

    4. Continuity

    5. Rules of differentiation

    6. Product and quotient rule

    7. Derivatives of trigonometric functions

    8. Chain rule

    9. Implicit differentiation

    10. Derivatives of logarithmic and exponential functions

    11. Logarithmic differentiation

    12. Derivatives of inverse trigonometric functions

    13. Maxima and minima

    14. What derivatives tell us

    15. Graphing Functions



  • Topic 1


    Calculus was first invented to meet the mathematical needs of scientists of the sixteenth and seventeenth centuries, needs that mainly mechanical in nature. Nowadays it is a tool used almost everywhere in the modern world to describe change and motion. Its use is widespread in science, engineering, medicine, business, industry, and many other fields. Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.


    The objective of this course is to introduce the fundamental ideas of the differential and integral calculus of functions of one variable.




    Limits and continuity. Derivatives, Rules of differentiation, Higher order derivatives, Chain rule. Related rates. Rolle's and the mean value theorem. Critical Points. Asymptotes. Curve sketching. Integrals, Fundamental Theorem, Techniques of integraion, Definite integrals. Application to geometry and science. Indeterminate forms. L'Hospital's Rule. Improper integrals. Polar Coordinates.



    This course is the first of a series of engineering mathematics courses. It is a prerequisite to Math152 - Calculus II, and to Math203 Ordinary Differential Equations (or similar courses on differential equations).



    On succesful completion of the course, the students should be able to:

    • recognise properties of functions and their inverses;
    • recall and use properties of polynomials, rational functions, exponential, logarithmic, trigonometric and inverse-trigonometric functions;
    • understand the terms domain and range;
    • sketch graphs, using function, its first derivative, and the second derivative;
    • use the algebra of limits, and l’Hôpital’s rule to determine limits of simple expressions;
    • apply the procedures of differentiation accurately, including implicit and logarithmic differentiation;
    • apply the differentiation procedures to solve related rates and extreme value problems;
    • obtain the linear approximations of functions and to approximate the values of functions;
    • perform accurately definite and indefinite integration, using parts, substitution, inverse substitution;
    • understand and apply the procedures for integrating rational functions;
    • perform accurately improper integrals;
    • calculate the volumes of solid objects, the length of arcs and the surface area;
    • perform polar-to-rectangular and rectangular-to-polar conversions.
    • Topic 2

      Limits and Continuity

      Section 2.1-2.2: The Idea of Limits, Definitions of Limits (1 hours)
      Section 2.3: Techniques for Computing Limits (1 hour)
      Section 2.4: Infinite Limits (1 hour)
      Section 2.5: Limits at Infinity (1 hour)
      Section 2.6: Continuity (1 hour)

    • Topic 3


      Section 3.1: Introducing the Derivative (1 hour)
      Section 3.2: Rules of Differentiation (1 hour)
      Section 3.3: Product and Quotient Rule (1 hour)
      Section 3.4: Derivatives of Trigonometric Functions (1 hour)
      Section 3.6: Chain Rule (1 hour)
      Section 3.7: Implicit Differentiation (1 hour)
      Section 3.8: Derivatives of Logarithmic and Exponential Functions (1hour)
      Section 3.9: Derivatives of inverse trigonometric Functions (1 hour)

    • Topic 4

      Application of the Derivative

      Section 4.1: Maxima and Minima (1 hour)
      Section 4.2: What Derivatives Tell us (2 hours)
      Section 4.3: Graphing Functions (2 hours)
      Section 4.5: Linear Approximations (1 hour)
      Section 4.6: The Mean Value Theorem (1 hour)
      Section 4.7: L’Hopital’s Rule (2 hours)
      Section 4.8: Antiderivatives (1 hour)


    • Topic 5


      Section 5.2: The Definite Integral (1hour)
      Section 5.3: Fundamental Theorem of Calculus (0.5 hour)
      Section 5.4: Working with Integrals (1 hour)
      Section 5.5: Substitution Rule (1.5 hours)
      Section 5.3: Fundamental Theorem of Calculus (0.5 hour)
      Section 5.4: Working with Integrals (1 hour)
      Section 5.5: Substitution Rule (1.5 hours)


    • Topic 6

      Application of Integration

      Section 6.2: Region between Curves (2 hours)
      • Topic 7

        Integration Techniques

        Section 7.1 Integration by Parts (2 hours)
        Section 7.2 Trigonometric Integrals (2 hours)
        Section 7.3 Trigonometric Substitution (2 hours)
        Section 7.4 Partial Fractions (2 hours)
        Section 7.7 Improper Integrals (2 hours)
      • Topic 8

        Sequences and Infinite Series

        Section 8.1-8.2: Sequences (2 hours)
        Section 8.3: Infinite Series (2 hours)
        Section 8.4: Divergence and Integral Test (1 hour)
        Section 8.5: Ratio, Root, and Comparison Test (1,5 hours)
        Section 8.6: Alternating Series (1,5 hours)