COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics Syllabus

Mathematics - Calculus, Coordinate Geometry and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics - Calculus, Coordinate Geometry and Statistics in COMEDK UGET, not a summary of it.

1. 1

   Limits, Continuity and Differentiation

   3 topics

   • Limits and Continuity
     • Evaluation of limits
     • Continuity and differentiability
   • Differentiation
     • Derivatives of standard functions
     • Chain rule, product and quotient rules
     • Implicit and logarithmic differentiation
   • Applications of Derivatives
     • Rate of change and tangents and normals
     • Increasing and decreasing functions
     • Maxima and minima
2. 2

   Integral Calculus

   3 topics

   • Indefinite Integration
     • Standard integrals and substitution
     • Integration by parts
     • Integration by partial fractions
   • Definite Integration
     • Properties of definite integrals
     • Definite integral as a limit of sum
   • Applications of Integrals
     • Area under simple curves
     • Area between two curves
3. 3

   Differential Equations and Vectors

   3 topics

   • Differential Equations
     • Order and degree
     • Variable separable equations
     • Linear differential equations
   • Vector Algebra
     • Addition and scalar multiplication
     • Dot and cross products
     • Scalar and vector triple products
   • Applications of Vectors
     • Area of triangle and parallelogram
     • Coplanarity and geometric applications
4. 4

   Coordinate Geometry

   4 topics

   • Straight Lines
     • Slope and forms of a line
     • Distance and angle between lines
     • Family of lines
   • Circles
     • Equation of a circle
     • Tangents and normals
   • Conic Sections
     • Parabola
     • Ellipse
     • Hyperbola
   • Three-Dimensional Geometry
     • Direction cosines and ratios
     • Equation of a line in space
     • Equation of a plane and distance
5. 5

   Probability and Statistics

   3 topics

   • Probability
     • Addition and multiplication theorems
     • Conditional probability and Bayes' theorem
     • Random variables and binomial distribution
   • Statistics
     • Measures of central tendency
     • Mean deviation, variance and standard deviation
   • Linear Programming
     • Formulation of LPP
     • Graphical solution of LPP

Mathematics - Calculus, Coordinate Geometry and Statistics flashcards for COMEDK UGET

18 of 50 cards from the Mathematics - Calculus, Coordinate Geometry and Statistics deck — real questions with worked answers.

1. State the definition of the limit of a function f(x) as x approaches a (epsilon-delta).

   lim(x→a) f(x) = L means: for every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε.

2. What three conditions must hold for a function f to be continuous at x = a?

   (1) f(a) is defined, (2) lim(x→a) f(x) exists, (3) lim(x→a) f(x) = f(a).

3. State the value of lim(x→0) (sin x)/x and lim(x→0) (tan x)/x.

   Both limits equal 1 (with x in radians).

4. What is lim(x→0) (1 − cos x)/x²?

   1/2.

5. Give the standard limits lim(x→0) (a^x − 1)/x and lim(x→0) (e^x − 1)/x.

   (a^x − 1)/x → ln a, and (e^x − 1)/x → 1.

6. State the first principle (definition) of the derivative of f at x.

   f'(x) = lim(h→0) [f(x+h) − f(x)] / h.

7. State the product rule and quotient rule for differentiation.

   Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

8. State the chain rule for differentiating a composite function y = f(g(x)).

   dy/dx = f'(g(x))·g'(x), i.e. dy/dx = (dy/du)(du/dx).

9. Give the derivatives of sin x, cos x, tan x, and sec x.

   d/dx sin x = cos x; cos x → −sin x; tan x → sec²x; sec x → sec x tan x.

10. Give the derivatives of e^x, a^x, ln x, and log_a x.

    e^x → e^x; a^x → a^x ln a; ln x → 1/x; log_a x → 1/(x ln a).

11. Give the derivatives of arcsin x, arctan x, and arcsec x.

    arcsin x → 1/√(1−x²); arctan x → 1/(1+x²); arcsec x → 1/(|x|√(x²−1)).

12. What is the geometric meaning of the derivative f'(a)?

    It is the slope of the tangent line to y=f(x) at x=a (instantaneous rate of change).

13. State the equations of the tangent and normal to y = f(x) at point (x₁, y₁).

    Tangent: y − y₁ = f'(x₁)(x − x₁). Normal: y − y₁ = −1/f'(x₁) · (x − x₁).

14. What is the second-derivative test for a local extremum at a critical point c?

    If f'(c)=0: f''(c)>0 ⇒ local minimum; f''(c)<0 ⇒ local maximum; f''(c)=0 ⇒ test inconclusive.

15. Define a point of inflection in terms of the second derivative.

    A point where the concavity changes; typically f''(x)=0 (or undefined) and f'' changes sign there.

16. When is a function increasing or decreasing on an interval?

    Increasing where f'(x) > 0; decreasing where f'(x) < 0 on that interval.

17. State Rolle's Theorem.

    If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

18. State the Mean Value Theorem (Lagrange).

    If f is continuous on [a,b] and differentiable on (a,b), then ∃ c in (a,b) with f'(c) = [f(b)−f(a)]/(b−a).

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