## DERIVATIVES

#### First Principle

# MATHEMATICS SYLLABUS

**1. Algebra**

**1.1 Logic and Set:** introduction of Logic, statements, logical connectives, truth tables, basic laws of logic, theorems based on set operations.

**1.2 Real numbers: **field axioms, order axioms, interval, absolute value, geometric representation of real numbers.

**1.3 Function: **Review, domain & range of a function, Inverse function, composite function, functions of special type, algebraic (linear, quadratic & cubic), Trigonometric, exponential, logarithmic)

**1.4 Curve sketching: **odd and even functions, periodicity of a function, symmetry (about origin, x-and y-axis), monotonicity of a function, sketching graphs of polynomials and some rational functions, Trigonometric, exponential, logarithmic function (simple cases only)

**1.5 Sequence and series: **arithmetic, geometric, harmonic sequences and series and their properties A.M, G.M, H.M and their relations, sum of infinite geometric series. 1.6 Matrices and determinants: Transpose of a matrix and its properties, Minors and cofactors, Adjoint, Inverse matrix, Determinant of a square matrix, Properties of determinants (without proof)

**1.7 Complex number: **definition imaginary unit, algebra of complex numbers, geometric representation, absolute value (Modulus) and conjugate of a complex numbers and their properties, square root of a complex number, polar form of complex numbers.

**2. Trigonometry**

**2.1 Properties of a triangle** (Sine law, Cosine law, tangent law, Projection laws, Half angle laws).

**2.2 Solution of triangle**(simple cases)

**3. Analytic Geometry**

**3.1 Straight Line: **length of perpendicular from a given point to a given line. Bisectors of the angles between two straight lines. Pair of straight lines: General equation of second degree in x and y, condition for representing a pair of lines. Homogenous second-degree equation in x and y. angle between pair of lines. Bisectors of the angles between pair of lines.

**3.2 Circle: **Condition of tangency of a line at a point to the circle, Tangent and normal to a circle.

**3.3 Conic section: ** Standard equation of parabola, equations of tangent and normal to a parabola at a given point.

**4. Vectors**

**4.1 Vectors: **collinear and non collinear vectors, coplanar and noncoplanar vectors, linear combination of vectors,

**4.2 Product of vectors: **scalar product of two vectors, angle between two vectors, geometric interpretation of scalar product, properties of scalar product, condition of perpendicularity.

**5. Statistics and Probability**

**5.1 Measure of Dispersion:** introduction, standard deviation), variance, coefficient of variation, Skewness (Karl Pearson and Bowley)

**5.2 Probability: **independent cases, mathematical and empirical definition of probability, two basic laws of probability (without proof).

**6. Calculus**

**6.1 Limits and continuity:** limits of a function, indeterminate forms. algebraic properties of limits (without proof), Basic theorems on limits of algebraic, trigonometric, exponential and logarithmic functions, continuity of a function, types of discontinuity, graphs of discontinuous function.

**6.2 Derivatives: **derivative of a function, derivatives of algebraic, trigonometric, exponential and logarithmic functions by definition (simple forms), rules of differentiation. derivatives of parametric and implicit functions, higher order derivatives, geometric interpretation of derivative, monotonicity of a function, interval of monotonicity, extreme of a function, concavity, points of inflection, derivative as rate of measure.

**6.3 Anti-derivatives: **anti-derivative. integration using basic integrals, integration by substitution and by parts methods, the definite integral, the definite integral as an area under the given curve, area between two curves.

**7. Computational Methods**

**7.1 Computing Roots: **Approximation & error in computation of roots in nonlinear equation, Algebraic and transcendental equations & their solution by bisection and Newton- Raphson Methods

**8. Mechanics or Mathematics for economics and Finance**

**8.1 Statics:** Forces and resultant forces, parallelogram law of forces, composition and resolution of forces, Resultant of coplanar forces acting on a point, Triangle law of forces and Lami's theorem.

**8.2 Dynamics: **Motion of particle in a straight line, Motion with uniform acceleration, motion under the gravity, motion down a smooth inclined plane. The concepts and theorem restated and formulated as application of calculus

**8.3 Mathematics for economics and finance:**

Mathematical Models and Functions, Demand and supply, Cost, Revenue, and profit functions, Elasticity of demand, supply and income , Budget and Cost Constraints, Equilibrium and break even

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