ICS (Intermediate in Computer Science) Mathematics (Part II) - Calculus and Analytic Geometry Syllabus

Mathematics (Part II) - Calculus and Analytic Geometry syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (Part II) - Calculus and Analytic Geometry in ICS (Intermediate in Computer Science), not a summary of it.

1. 1

   Functions and Limits

   4 topics

   • Types of Functions
   • Limit of a Function
   • Theorems on Limits
   • Continuity and Discontinuity
2. 2

   Differentiation

   4 topics

   • Derivative by First Principle
   • Rules of Differentiation
   • Chain Rule and Implicit Differentiation
   • Derivatives of Trigonometric, Exponential and Logarithmic Functions
3. 3

   Integration

   4 topics

   • Indefinite Integration
   • Methods of Integration
   • Definite Integration
   • Area Under a Curve
4. 4

   Introduction to Analytic Geometry

   3 topics

   • Coordinate System and Distance Formula
   • Division of a Line Segment
   • Translation and Rotation of Axes
5. 5

   Linear Inequalities and Linear Programming

   3 topics

   • Linear Inequalities in Two Variables
   • Feasible Region
   • Optimization and Objective Function
6. 6

   Conic Sections

   4 topics

   • The Circle
   • The Parabola
   • The Ellipse
   • The Hyperbola
7. 7

   Vectors

   4 topics

   • Introduction to Vectors
   • Scalar (Dot) Product
   • Vector (Cross) Product
   • Scalar Triple Product

Mathematics (Part II) - Calculus and Analytic Geometry flashcards for ICS (Intermediate in Computer Science)

20 of 50 cards from the Mathematics (Part II) - Calculus and Analytic Geometry deck — real questions with worked answers.

1. Define a function from set X to set Y.

   A function f: X to Y is a rule that assigns to each element x in X exactly one element y in Y. X is the domain, Y is the codomain, and the set of all images is the range.

2. What is an even function and what symmetry does its graph have?

   A function f is even if f(-x) = f(x) for all x in the domain. Its graph is symmetric about the y-axis. Example: f(x) = x^2, cos x.

3. What is an odd function and what symmetry does its graph have?

   A function f is odd if f(-x) = -f(x) for all x in the domain. Its graph is symmetric about the origin. Example: f(x) = x^3, sin x.

4. Distinguish between an algebraic function and a transcendental function.

   An algebraic function involves only algebraic operations (polynomial, rational, radical) on x. A transcendental function cannot be expressed algebraically; examples include trigonometric, inverse trigonometric, exponential and logarithmic functions.

5. What is a one-to-one (injective) function?

   A function f is one-to-one if distinct inputs give distinct outputs: f(x1) = f(x2) implies x1 = x2. Graphically, it passes the horizontal line test.

6. Define the composition of functions (fog)(x).

   (fog)(x) = f(g(x)). The output of g becomes the input of f. It is defined where g(x) lies in the domain of f. In general fog is not equal to gof.

7. What condition must a function satisfy to have an inverse, and how are domain and range related?

   A function must be one-to-one (bijective on its range) to have an inverse f^-1. The domain of f becomes the range of f^-1, and the range of f becomes the domain of f^-1.

8. Give the definitions of the hyperbolic functions sinh x and cosh x.

   sinh x = (e^x - e^-x)/2 and cosh x = (e^x + e^-x)/2. They satisfy the identity cosh^2 x - sinh^2 x = 1.

9. Give the intuitive meaning of the limit of a function: lim(x->a) f(x) = L.

   It means f(x) can be made as close to L as we wish by taking x sufficiently close to a (but not equal to a). The behaviour of f exactly at a is irrelevant.

10. State the formal (epsilon-delta) definition of lim(x->a) f(x) = L.

    For every epsilon > 0 there exists delta > 0 such that whenever 0 < |x - a| < delta, it follows that |f(x) - L| < epsilon.

11. What is the relationship between one-sided limits and the existence of a limit?

    lim(x->a) f(x) = L exists if and only if both one-sided limits exist and are equal: lim(x->a-) f(x) = lim(x->a+) f(x) = L.

12. Evaluate lim(x->0) (sin x)/x.

    lim(x->0) (sin x)/x = 1, where x is measured in radians. This is a fundamental trigonometric limit.

13. Evaluate lim(x->0) (1 - cos x)/x.

    lim(x->0) (1 - cos x)/x = 0. (A related limit is lim(x->0) (1 - cos x)/x^2 = 1/2.)

14. Evaluate lim(x->0) (a^x - 1)/x.

    lim(x->0) (a^x - 1)/x = ln a. In particular, lim(x->0) (e^x - 1)/x = 1.

15. State the two standard limits that define the number e.

    e = lim(n->infinity) (1 + 1/n)^n, and equivalently e = lim(x->0) (1 + x)^(1/x). The value is approximately 2.71828.

16. Evaluate lim(x->infinity) (1 + 1/x)^x.

    lim(x->infinity) (1 + 1/x)^x = e (approximately 2.71828).

17. State the limit laws for sum and product of functions.

    If lim f(x) = L and lim g(x) = M, then lim [f(x) + g(x)] = L + M and lim [f(x) g(x)] = L M (limits taken as x->a).

18. State the quotient law for limits and its condition.

    If lim f(x) = L and lim g(x) = M with M not equal to 0, then lim [f(x)/g(x)] = L/M (as x->a).

19. State the Sandwich (Squeeze) Theorem for limits.

    If g(x) <= f(x) <= h(x) near a, and lim(x->a) g(x) = lim(x->a) h(x) = L, then lim(x->a) f(x) = L.

20. How do you evaluate lim(x->a) of a polynomial function?

    For a polynomial p(x), lim(x->a) p(x) = p(a) by direct substitution, because polynomials are continuous everywhere.

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