Syllabus of USMT101201

Prerequisites

1. Limits of some standard functions as [math]x[/math] approaches [math]a[/math] ([math]a\in \mathbb{R}[/math]), such as constant function, [math]x^n[/math], [math]\sin x[/math], [math]\cos x[/math], [math]\tan x[/math], exponential and logarithmic functions,[math]\lim\limits_{x\rightarrow 0}\frac{\sin x}{x}[/math], continuity in terms of limits.
2. Derivatives, Derivatives of standard functions such as constant function, [math]x^n[/math], trigonometric functions, [math]e^x, a^x[/math] (a > 0), [math]\log x[/math]

Unit 1 : Limit and continuity of functions of one variable

(a)

• Absolute value of a real number and the properties such as : [math]|-a|=|a|, |ab|=|a||b|[/math], and [math]|a+b|\leq|a|+|b|[/math]
• Intervals in [math]\mathbb{R}[/math], neighbourhoods and deleted neighbourhoods of a real number , bounded subsets of [math]\mathbb{R}[/math].

(b)

• Graphs of functions such as [math]|x|, \frac{1}{x}, ax^2+bx+c, \lceil x\rceil, \lfloor x\rfloor, x^n, \sin x, \cos x, \tan x, \sin\frac{1}{x}, x\sin\frac{1}{x}[/math] over suitable intervals.
• Graph of an objective function and its inverse. Examples such as [math]x^2[/math] and [math]x^{1/2}[/math], [math]x^3[/math] and [math]x^{1/3}, ax+b (a\neq 0)[/math] and [math]\frac{x}{a}-\frac{b}{a}[/math] over suitable domains.

(c)

• Statement of rules for finding limits ,sum rule, difference rule , product rule , constant multiple rule , quotient rule
• Sandwich theorem of limits (without proof)
• Limit of composite functions (without proof).

(d)

• [math]\epsilon-\delta[/math] definition of limit of a real valued function, simple illustrations like [math]ax+b, \sqrt{x+a}, a\gt0, x^2, \sin x, \cos x[/math].(In general, evaluation of limits to be done using rules in (c) )
• Formal definition of infinite limits. examples such as [math]\lim\limits_{x\rightarrow 0}\frac{1}{x}[/math]

(e)

• Continuity of a real valued function at point in terms of limits, and two sided limits. Graphical representation of continuity of a real valued function.
• Continuity of a real valued function at end points of domain.
• Removable discontinuity at a point of a real valued function and extension of a function having removable discontinuity at a point to a function continuous at that point.
• Continuity of polynomials and rational functions.
• Constructing a real valued function having finitely many prescribed points of discontinuity over an interval.
• Continuity of a real valued function over an interval. Statements of properties of continuous functions such as the following:
• Intermediate value property.
• A continuous function on a closed and bounded interval is bounded and attains its bounds.
• Elementary consequences such as if [math]f:[a,b]\rightarrow \mathbb{R}[/math] is continuous then range of [math]f[/math] is a closed and bounded interval.

(g)

• Definition of limit as [math]x[/math] approaches [math]\pm\infty[/math] examples. Limits of rational functions as [math]x[/math] approaches [math]\pm\infty[/math].

Unit 2 : Differentiability of functions of one variable

(a) Definition of derivative of a real valued function at a point. notion of differentiability, geometric interpretation of a derivative of a real valued function at a point, differentiability of a function over an interval, statement of rules of differentiability, chain rule of finding derivative of composite differentiable functions, derivatives of an inverse function (without proof) and its applications, implicit differentiation.
(b)

• Differentiable functions are continuous, but the converse is not true.
• Higher order derivatives, examples of functions [math]x^n|x|, n = 0, 1, 2 \ldots[/math] which are differentiable [math]n[/math] times but not [math]n+1[/math] times.
• Leibnitz Theorem for [math]n^{th}[/math] order derivative of product of two [math]n[/math] times differentiable functions.

Unit 3 : Applications of derivatives

(a)

• Mean Value Theorems: Rolle's Mean Value Theorem, Lagrange's Mean Value Theorem, Cauchy 's Mean Value Theorem.

(b) Extreme values of functions, absolute and local extrema, critical points, increasing and decreasing functions, the second derivative test for extreme values.

(c) Graphing of functions using first and second derivatives, the second derivative test for concavity, points of inflection.

(d) Asymptotes- horizontal and vertical.

Unit 4 : Analytic Geometry in Euclidean spaces

(a) Review of vectors in [math]\mathbb{R}^2, \mathbb{R}^3[/math] component form of vectors, basic notions such as addition and scalar multiplication of vectors, dot product of vectors, orthogonal vectors, length (norm) of a vector, unit vector, distance between two vectors, cross product of vectors in [math]\mathbb{R}^3[/math] scalar triple product (box product), vector projections.

(b) Lines and planes in space, equation of sphere, cylinders and quadric surfaces.

(c) Polar co-ordinates in [math]\mathbb{R}^2[/math] polar graphing with examples: [math]r=\sin\theta\;, r=\cos 2\theta,\; r=a(1-\cos\theta)[/math]

(d) Relationship between polar and Cartesian co-ordinates in [math]\mathbb{R}^2[/math] cylindrical and spherical co-ordinates in [math]\mathbb{R}^3[/math] and relationships of these co-ordinates with Cartesian co-ordinates and each other.

Unit 5 : Limits and continuity of functions of two and three variables

(a)

• Open disc in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math], boundary of open disc, closed disc in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math], bounded regions, unbounded regions in [math]\mathbb{R}^2[/math] and [math]\mathbb{R}^3[/math].
• Real valued functions of two or three variables, examples. Level curves for functions of two variables. Use of level curves to draw graphs of z = ƒ (x,y), especially quadric surfaces, [math]\epsilon-\delta[/math] definition of a limit of a real valued function of two variables (only brief statement ).

(b) Statement of rules of limits in two (or three) variables; Sum rule, difference rule, product rule, constant multiple rule, quotient rule, power rule.

• Applying these rules to determine limits of polynomial and rational functions.
• Definition of continuity of functions of two (or three) variables in terms of limits.

(c) Definition of a path. Limit of a function along paths. Two path test for non-existence of a limit. Examples of functions such as: [math]\frac{2xy}{x^2+y^2}, \frac{2x^2y}{x^4+y^2}, \frac{-xy}{x^2+y^2}[/math] etc.Sandwich theorem for a function of two variables. (without proof).

(d) Calculating [math]\lim\limits_{(x,y)\rightarrow (0,0)}f(x,y)[/math] by changing to polar co-ordinates (illustrating with examples).

(e) Vector valued functions of one and several variables, planar and space curves, component functions, vector fields, graphs of vector valued functions like [math](\cos t, \sin t), (\cos t, \sin t, 1), (\cos t, \sin t, t)[/math]. Limits of vector valued function by taking limits of component functions.

Unit 6 : Differentiability of functions of two variables

(a)

• Partial derivatives of a real valued function of two variables, the relationship between continuity and the existence of partial derivatives at a point. Second order partial derivatives, Mixed derivative theorem for two variables (without proof). The increment theorem for two variables (without proof).
• Differentiability of a function of two variables at a point over a disc, linearization of a differentiable function at a point.

• Chain rule for composite function of the type [math]\mathbb{R}^2\xrightarrow{f}\mathbb{R}\xrightarrow{g}\mathbb{R}[/math] (without proof).

• Implicit differentiation.

(b) Directional derivatives in a plane, interpretation of directional derivatives, gradient vector, relation between directional derivative and gradient.

(c) Geometric interpretation of partial derivatives and its relation to the tangent plane at a point.

(d) Extreme values of a function of two variables. Local maximum, local minimum and first derivative test for local extreme values (without proof). Critical points, saddle points, second derivative test for local extreme values (without proof).

(e) The method of Lagrange’s Multiplier to obtain extrema of a function of two variables (one constraint only).

(f) Derivatives of vector valued function as derivative of component functions, Statement of rules of differentiation - sum, difference, product, constant multiple. Chain rule for composite function of the type [math]\mathbb{R}^2\xrightarrow{f}\mathbb{R}\xrightarrow{g}\mathbb{R}[/math] (without proof). Geometric interpretation of derivatives. Derivatives of dot and cross products.

Recommended Reading

1. G.B. Thomas and R.L. Finney, Calculus and Analytic Geometry, Ninth Edition, Addison - Wesley, 1998.
2. Sudhir R. Ghorpade and Balmohan V. Limaye, A course in Calculus and Real Analysis, Springer International Edition.
3. Howard Anton, Calculus - A new Horizon, Sixth Edition, John Wiley and Sons Inc. 1999.
4. James Stewart, Calculus, Third Edition, Brooks / cole Publishing Company, 1994.