Real Analysis & Calculus

• Basics & Real Line:
  • [[Limits]], [[continuity]], [[intermediate value property]], [[sequence and limit]].
• Sets & Relations:
  • [[Countable and uncountable sets]].
  • [[Equivalence relations]] and [[partitions]].
  • [[Elementary set theory]], [[relations]], [[functions]], [[binary operations]].
• Sequences & Series (Numbers):
  • [[Convergence and divergence of sequences and series]].
  • [[Bounded and monotone sequences]].
  • [[Cauchy sequences and completeness]].
  • [[Bolzano-Weierstrass theorem]].
  • [[Absolute convergence]], [[tests of convergence]] (comparison, ratio, root).
  • [[Limsup, liminf]].
• Power Series:
  • [[Radius and interval of convergence]].
  • [[Term-wise differentiation and integration of power series]].
  • [[Taylor series and Maclaurin series]].
  • [[Taylor expansions]].
• Differentiation (One Variable):
  • [[Differentiability]].
  • [[Rolle’s theorem]] and [[mean value theorem]].
  • [[L’Hospital rule]], [[Taylor’s theorem]].
  • [[Maxima and minima]].
• Integration (One Variable):
  • [[Riemann integration]] (definite integrals and properties, existence of Riemann integral).
  • [[Fundamental theorem of calculus]].
  • [[Change of variable]], [[improper integrals]].
• Multivariable Calculus:
  • [[Functions of two, three, or several variables]].
  • [[Limits and continuity]].
  • [[Directional derivatives]], [[partial derivatives]].
  • [[Total derivative]].
  • [[Maxima and minima]].
  • [[Saddle point]], [[method of Lagrange’s multipliers]].
  • [[Chain rule]].
  • [[Jacobian]].
  • [[Differentiation of functions from R to Rn]].
  • [[Inverse and Implicit function theorems]].
• Multiple Integrals:
  • [[Double and triple integrals]].
  • [[Area, volume, and surface area using multiple integrals]].
  • [[Change of order of integration]].
• Vector Calculus:
  • [[Gradient]], [[divergence]], and [[curl]].
  • [[Line integrals]] and [[surface integrals]].
  • [[Vector fields]].
  • [[Green’s theorem]], [[Stokes’ theorem]], and [[Gauss divergence theorem]].
• Sequences & Series of Functions:
  • [[Pointwise and uniform convergence]].
  • [[Uniform continuity]].
  • [[Ascoli-Arzela theorem]].
  • [[Weierstrass approximation theorem]].
• Lebesgue Measure:
  • [[Lebesgue measure on the real line]], [[measurable functions]].
  • [[Lebesgue integral]], [[Fatou’s lemma]], [[monotone convergence theorem]], [[dominated convergence theorem]].

One Variable

Limit

Continuity A function f(x) is said to be continuous at a point x=a, if following conditions are satisfied: 1. f(a) exists 2. \displaystyle\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = \lim_{x\to a}f(x) = f(a)

Differentiability A function is said to be differentiable at x=a if left hand derivative (LHD) and right hand derivative (RHD) are equal. i.e. \lim_{h\to 0}\frac{f(x-h)-f(x)}{-h}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

L’Hospital rule of limits the limit of the function \displaystyle\frac{f(x)}{g(x)}, where g(x)\neq 0, can be calculated by \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)} - L’Hospital rule can only be applied if the limit of the function gives indeterminate form.

Cauchy’s definition of limits A real values function f defined on domain D, is said to have a limit at x=c if \exists\epsilon,\delta >0:\forall x\in D and 0<|x-c|<\delta |f(C)-l|<\epsilon

Borel’s Theorem If f is a continuous function in the closed interval [a,b], then this interval can always be divided into finite number of subintervals such that, for \epsilon>0, |f(x_1)-f(x_2)|<\epsilon, where x_1 and x_2 are any two points inside the subinterval.

Boundedness Theorem If a function f is continuous in the closed interval [a,b] then f is bounded in closed interval [a,b].

Bolzano’s Theorem If f is a continuous function in the closed interval [a,b] and f(a) and f(b) are of opposite sides, then there exists at least one value of x in this interval for which f(x) vanishes, i.e. f(x)=0.

Intermediate Value Theorem If a function f is continuous in [a,b] and f(a)\neq f(b), then f(x) must take, at least once, all the values between a and b.

Extreme Value Theorem If a function f\in C in [a,b], then it attains its maxima and minima in [a,b].

Darboux’s Theorem - Intermediate Value Theorem for Derivatives If f is finitely differentiable function in [a,b] and, f'(a) and f'(b) are of opposite signs, then there exists at least one point c\in(a,b) such that, f'(c)=0.

Lagrange’s Mean Value Theorem If f(x) is a function such that it is continuous in [a,b] and differentiable in (a,b), then \exists c\in(a,b): f'(c)=\frac{f(b)-f(a)}{b-a}

Geometrical interpretation The Lagrange’s mean value theorem states that for any given curve there exists a point at which the slope of the tangent to the curve is same as slope of line joining the end points of that curve.

Rolle’s Theorem (Special case of Lagrange’s MVT) If f(x) is a function such that it is continuous in [a,b], differentiable in (a,b) and f(a)=f(b), then \exists c\in(a,b): f'(c)=0

Cauchy’s Mean Value Theorem Let f and g are two functions defined in [a,b] such that f and g are continuous in [a,b] and differentiable in (a,b). Then, \exists at least one c \in (a,b): \frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}

Successive differentiation

y	y_n (nth derivative)
e^ax	a^ne^{ax}
\sin(ax+b)	a^{n}\sin{\left(\frac{n\pi}{2}+ax+b\right)}
(ax+b)^{m}	a^{n} \frac{m!}{(m-n)!} (ax+b)^{m-n}
if m=n	a^{n}n!
if n>m	0
if m=-r	a^{n}(-1)^{n} \frac{(r+n-1)!}{(r-1)!}\frac{1}{(ax+b)^{n+r}}
e^{ax}\sin{(bx+c)}	\sqrt{(a^{2}+b^{2})^{n}}e^{ax}\sin{\left(n\tan^{-1}{\left(\frac{b}{a}\right)}+bx+c \right)}

Leibnitz’s Theorem Let u and v be two functions of x. Then the n^{th} derivative of uv is, (uv)_{n}=\binom{n}{0}u_{n}v+\binom{n}{1}u_{n-1}v_{1}+\binom{n}{2}u_{n-2}v_2+\cdots+\binom{n}{n}uv_{n} where u_n,v_n are the n^{th} derivatives. - consider v as the function whose derivative will become 0 after a few differentiations

Taylor’s Theorem

Taylor’s series If f(a+h) can be expanded in terms of a series of ascending powers of h, then f(a+h)=f(a)+hf'(a)+\frac{h^{2}f''(a)}{2!}+\cdots Case 1 If h=x-a, f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^{2}f''(a)}{2!}+\cdots Case 2 (Maclaurin’s series) If a=0 in Case 1, f(x)=f(0)+xf'(0)+\frac{x^{2}f''(0)}{2!}+\cdots

Partial derivatives For z=f(x,y), first order partial derivatives are \frac{\delta z}{\delta x}=f_{x},\, \frac{\delta z}{\delta y}=f_{y} and second order partial derivatives are \frac{\delta^{2} z}{\delta x^{2}}=f_{xx},\, \frac{\delta^{2} z}{\delta y^{2}}=f_{yy},\, \frac{\delta^{2} z}{\delta x\delta y}=f_{xy},\, \frac{\delta^{2} z}{\delta y\delta x}=f_{yx}

Homogeneous function every term is of the same degree f(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+a_{2}x^{n+2}y^{2}+\cdots+a_{n-1}xy^{n-1}+a_{n}y^{n} is a homogeneous function of x and y of degree n. This can also be written as f(x,y)=x^{n}\left\{a_{0}+a_{1}\left(\frac{y}{x}\right)+a_{2}\left(\frac{y}{x}\right)^{2}+\cdots+a_{n-1}\left(\frac{y}{x}\right)^{n-1}+a_{n}\left(\frac{y}{x}\right)^{n}\right\} or \displaystyle x^{n}f\left(\frac{y}{x}\right), where \displaystyle f\left(\frac{y}{x}\right) is some function of \displaystyle\left(\frac{y}{x}\right).

Euler’s Theorem If u(x,y) is a homogeneous function of degree n, then x\frac{\delta u}{\delta x}+y\frac{\delta u}{\delta y}=nu

Tangents equation of a tangent at point A(x_{1},y_{1}) is given by, y-y_{1}=\frac{dy}{dx}(x-x_{1})

Normal line perpendicular to the tangent at the point of contact of the tangent. The equation of a normal at A(x_{1},y_{1}) is given by, y-y_{1}=-\frac{dx}{dy}(x-x_{1}) (for equation of normal, put dy/dx=-dx/dy in the equation of tangent).

f(x)	Condition	In words
Concavity in interval I	f''(x)>0\,\forall\,x\in I	if graph of f(x) is above all the tangents in I
Convexity in interval I	f''(x)<0\,\forall\,x\in I	if graph of f(x) is below all the tangents in I

Point of inflection x_0 is the point of inflection where the curve changes from being concave to convex or vice versa. Points of inflection are given by, f''(x)=0

Curvature (related to change of direction of a curve) It measures how sharply the bends in the curve occur. Curvature is given by k=\frac{1}{\rho} where \rho is the radius of curvature.

Radius of curvature	\rho=
If tangent is parallel to x-axis	\frac{(1+y'^{2})^{\frac{3}{2}}}{y''}
If tangent is parallel to y-axis	\frac{(1+x'^{2})^{\frac{3}{2}}}{x''}
For parametric curves	\frac{\left(x'^{2}+y'^{2}\right)^{\frac{3}{2}}}{x'y''-x''y'}

Asymptotes a line that approaches to the given curve as one or both x and y axis coordinates tend to infinity but never intersects or crosses the curve.

Rectangular asymptote If any asymptote is parallel to x or y axis, then it is called a rectangular asymptote. If parallel to - x-axis, its horizontal asymptote - y-axis, its vertical asymptote

Oblique asymptote any asymptote which is not rectangular

Asymptote	Working rule
Horizontal	put coeff. of highest power of x equal to 0
Vertical	put coeff. of highest power of y equal to 0
Oblique, y=mx+c	of the curve \phi_{n}(x,y)+\phi_{n-1}(x,y)+\cdots+\phi_{2}(x,y)+\phi_{1}(x,y)+\phi_{0}(x,y) where n is the degree of the terms. 1. Put x=1 and y=m in the above equation to get the terms \phi_{n}(1,m),\,\phi_{n-1}(1,m),\,\ldots,\,\phi_{1}(1,m),\,\phi_{0}(1,m) 2. Put \phi_{n}(1,m)=0 and obtain roots of m 3. Case 1 If roots are real and distinct, then c=-\frac{\phi_{n-1}(m)}{\phi'_{n}(m)} Case 2 If roots are real and equal, then \frac{c^{2}}{2!}\phi''_{n}(m)+c\phi'_{n-1}(m)+\phi_{n-2}(m)=0

Singular points A point in curve at which the curve exhibits an extra-ordinary behaviour. There are two types of singular points: 1. point of inflection 2. multiple points

Multiple points A point on the curve through which more than one branch of the curve pass is called a multiple point.

(Or) a multiple point of the curve f(x,y)=0 is the point (x,y) on the curve where more than one tangent can be drawn.

Multiple points of rth order Point at which r branches of the curve pass.

Double point multiple point of 2^{nd} order is called a double point. - two tangents in general can be drawn through a double point, they can be - real and different - real and coincident - imaginary

Type of double point	Meaning
Node	two real branches pass, two tangents, real and different
Cusp	two real branches pass, two tangents, real and coincident
Conjugate or isolated point	\exists no real points on the curve in the neighborhood of the point

working rule to determine double points on the curve f(x,y)=0: 1. obtain partial derivatives f_x,f_y,f_{xy},f_{xx},f_{yy} 2. put f_{x}=0, f_{y}=0 and get (x_{1},y_{1}) 3. if f(x_{1},y_{1})=0 \implies (x_{1},y_{1}) is a double point 4. if (f_{xy})^{2}-f_{xx}f_{yy} - >0\implies node - =0\implies cusp - <0\implies conjugate or isolated point

Maxima and minima Riemann integration (definite integrals and their properties) Fundamental theorem of calculus

Curve Tracing (SOAP method) - Symmetry: - if x is even \implies symmetric along y-axis - if y is even \implies symmetric along x-axis - if equation remains same after interchanging x and y, it is symmetric along both axes - Origin: if no constant OR put x=0,y=0, if equation is satisfied \implies curve passes through origin - Asymptotes: find rectangular and oblique asymptotes - POI: find point of inflection with the coordinate axes and the line of symmetry

Definite integrals as limit of sum \int_a^b f(x)dx=\lim_{n\to\infty} h[f(a)+f(a+h)+f(a+2h)+\cdots+f(a+(n-1)h)]

Fundamental Theorem of Integral Calculus If f is an Reimann integral function on [a,b] and F is a differentiable primitive function such that, F'(x)=f(x),a\leq x\leq b, then \int_a^b f(t)dt=F(b)-F(a)

Improper Integrals - 1st kind: integrand is continuous but limits are infinite A)\int_a^\infty=\lim_{b\to\infty}\int_a^b\quad B)\int_{-\infty}^b=\lim_{a\to{-\infty}}\int_a^b\quad C)\int_{-\infty}^\infty=\lim_{a\to{-\infty}}\int_a^c+\lim_{b\to\infty}\int_c^b - 2nd kind: limits are finite by integrand is infinite

2nd Kind
limits are finite by integrand is infinite
a) singular point at b