Class 6 Notes
24 sept 2018 log x. A more refined answer: it looks like a certain integral called ... Proof (Ben): The derivative of logx is 1 x and the derivative of.
LS
1 Theory of convex functions
1 mar 2016 Strongly convex if ∃α > 0 such that f(x) − α
ORF S Lec gh
Chapter 8 Logarithms and Exponentials: logx and e
Exercise 4 Prove the Laws of Exponents. Hint: make use of the fact that log x is a 1-1 function. Derivatives of the Exponential Function. We already
chapter
New sharp bounds for the logarithmic function
5 mar 2019 In this paper we present new sharp bounds for log(1 + x). We prove that our upper bound is sharper than all the upper bounds presented ...
Proof of the Sheldon Conjecture
13 feb 2019 x log x for all x ≥ 17. (1). January 2014]. PROOF OF THE SHELDON CONJECTURE ... Letting z = log x this derivative is z + log z + 1/z.
sheldon
Some Inequalities involving ( r!)1/ r
Lemma 1. If x> 1 then. 0<log (r(x))-{(x-$) log (x)-x+i log (2«). Proof. Proof. We prove that for JC^6 the second derivative of h(x) is negative.
div class title some inequalities involving span class italic r span span class sup span class italic r span span div
Chapter 3 Elementary Prime Number Theory
log p ps . Thus without justification of proving that the derivative of a series Denote the sum over the logarithm of primes by θ (x) = ∑ p≤x log p.
Notes
CONTINUITY AND DIFFERENTIABILITY
log. = x b. 6. logb b = 1 and logb 1 = 0. (iv) The derivative of ex Example 9 If ex + ey = ex+y prove that ... Example 13 If xy = ex–y
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1 Fisher Information
6 abr 2016 log f(X
Fisher
3. Convex functions
zk) (from Cauchy-Schwarz inequality) geometric mean: f(x)=(∏ n k=1 xk). 1/n on R n. ++ is concave. (similar proof as for log-sum-exp). Convex functions.
functions
5.1 Overview
5.1.1Continuity of a function at a point
Letf be a real function on a subset of the real numbers and letc be a point in thedomain off. Thenf is continuous atc iflim ( ) ( )x cf x f c®=More elaborately, if the left hand limit, right hand limit and the value of the function
atx =c exist and are equal to each other, i.e.,lim ( ) ( ) lim ( ) x cx cf x f c f x-+®®= =thenf is said to be continuous atx =c.5.1.2Continuity in an interval
(i)f is said to be continuous in an open interval (a,b) if it is continuous at every point in this interval. (ii)f is said to be continuous in the closed interval [a,b] if f is continuous in (a,b) lim x a+® f (x) =f (a) -lim x b® f (x) =f (b)Chapter 5CONTINUITY AND
DIFFERENTIABILITY
CONTINUITY AND DIFFERENTIABILITY 875.1.3Geometrical meaning of continuity (i) Functionf will be continuous atx =c if there is no break in the graph of the function at the point( ), ( )c f c. (ii) In an interval, function is said to be continuous if there is no break in the graph of the function in the entire interval.5.1.4Discontinuity
The functionf will be discontinuous atx =a in any of the following cases : (i)lim x a-® f (x) andlim x a+® f (x) exist but are not equal. (ii)lim x a-® f (x) andlim x a+® f (x) exist and are equal but not equal to f (a). (iii) f (a) is not defined.5.1.5Continuity of some of the common functions
Function f (x)Interval in which
f is continuous1. The constant function, i.e. f (x) =c
2. The identity function, i.e. f (x) =xR
3. The polynomial function, i.e.
f (x)= a0xn +a1xn-1 + ... +an-1x +an4. |x -a |(-¥,¥)
5.x-n,n is a positive integer(-¥,¥) - {0}
6.p (x) /q (x), wherep (x) andq (x) areR - {x :q (x) = 0}
polynomials inx7. sinx, cosxR
8. tanx, secxR- { (2n + 1)π
2:nÎZ}
9. cotx, cosecxR- { (np :nÎZ}
88 MATHEMATICS10.exR
11. logx(0,¥)
12. The inversetrigonometric functions,In their respective
i.e., sin -1x, cos-1x etc.domains5.1.6Continuity of composite functions
Letf andg be real valued functions such that (fog) is defined ata. Ifg is continuous ata andf is continuous atg(a), then (fog) is continuous ata.5.1.7Differentiability
The function defined byf¢(x) =0( ) ( )limhf x h f x h®+ -, wherever the limit exists, is
defined to be the derivative off atx. In other words, we say that a functionf is differentiable at a pointc in its domain if both0( ) ( )lim hf c h f c h -®+ -, called left hand derivative, denoted by L f¢ (c), and0( ) ( )lim hf c h f c h +®+ -, called right hand derivative, denoted by Rf¢ (c), are finite and equal. (i) The functiony =f (x) is said to be differentiable in an open interval (a, b) if it is differentiable at every point of (a, b) (ii) The functiony =f (x) is said to be differentiable in the closed interval [a, b] if Rf¢(a) and Lf¢ (b) exist andf¢ (x) exists for every point of (a,b). (iii) Every differentiable function is continuous, but the converse is not true5.1.8Algebra of derivatives
Ifu,v are functions ofx, then
(i)( )d u v d x±= ±du dv dx dx(ii)( )= +d dv duuv u vdx dx dx(iii)2du dv v ud udx dx dx v v-CONTINUITY AND DIFFERENTIABILITY 895.1.9 Chain rule is a rule to differentiate composition of functions. Letf =vou. If
t =u (x) and bothdt dx anddv dt exist then.=df dv dt dx dt dx5.1.10 Following are some of the standard derivatives (in appropriate domains) 1.-121(sin )1=-dxdxx2.-1
21(cos )1dxdxx-=-3.-1
21(tan )1=+dxdxx4.-1
21(cot )1dxdxx-=+5.-1
21(sec ), 11dxxdxx x=>-6.-1
21(cosec ), 11dxxdxx x-=>-5.1.11Exponential and logarithmic functions
(i) The exponential function with positive baseb> 1 is the function y =f (x) =bx. Its domain isR, the set of all real numbers and range is the set of all positive real numbers. Exponential function with base 10 is called the common exponential function and with basee is called the natural exponential function. (ii) Letb > 1 be a real number. Then we say logarithm ofa to baseb isx ifbx=a, Logarithm ofa to the baseb is denoted by logba. If the baseb = 10, we say it is common logarithm and ifb = e, then we say it is natural logarithms. logx denotes the logarithm function to basee. The domain of logarithm function isR+, the set of all positive real numbers and the range is the set of all real numbers. (iii) The properties of logarithmic function to any baseb> 1 are listed below:1. log
b (xy) = logbx + logby2. log
y = logbx - logby90 MATHEMATICS3. logbxn = nlogb x
4.logloglogc
b cx xb= , wherec > 15. log
bx1 log= xb6. logbb =1 and logb 1 = 0 (iv) The derivative ofexw.r.t.,x isex , i.e.( )x xde edx=. The derivative of logx w.r.t.,x is1 x; i.e.1(log )dxdx x=.5.1.12Logarithmic differentiation is a powerful technique to differentiate functions
of the formf (x) = (u (x))v(x), where bothf andu need to be positive functions for this technique to make sense.5.1.13Differentiation of a function with respect to another function
Letu =f (x) andv =g (x) be two functions ofx, then to find derivative of f (x) w.r.t. tog (x), i.e., to finddu dv, we use the formuladu dudx dvdv dx=.5.1.14Second order derivative2
2d dy d y
dx dx y2 , ify =f (x).
5.1.15Rolle"s Theorem
Letf : [a,b]®R be continuous on [a,b] and differentiable on (a,b), such thatf(a) =f (b), wherea andb are some real numbers. Then there exists at least one pointc in (a,b) such thatf¢ (c) = 0.CONTINUITY AND DIFFERENTIABILITY 91Geometrically Rolle"s theorem ensures that there is at least one point on the curve
y =f (x) at which tangent is parallel tox-axis (abscissa of the point lying in (a,b)).5.1.16Mean Value Theorem (Lagrange)
Letf : [a,b]®R be a continuous function on [a,b] and differentiable on (a,b). Then there exists at least one pointc in (a,b) such thatf¢ (c) =( ) ( )f b f a b a-5.1 Overview
5.1.1Continuity of a function at a point
Letf be a real function on a subset of the real numbers and letc be a point in thedomain off. Thenf is continuous atc iflim ( ) ( )x cf x f c®=More elaborately, if the left hand limit, right hand limit and the value of the function
atx =c exist and are equal to each other, i.e.,lim ( ) ( ) lim ( ) x cx cf x f c f x-+®®= =thenf is said to be continuous atx =c.5.1.2Continuity in an interval
(i)f is said to be continuous in an open interval (a,b) if it is continuous at every point in this interval. (ii)f is said to be continuous in the closed interval [a,b] if f is continuous in (a,b) lim x a+® f (x) =f (a) -lim x b® f (x) =f (b)Chapter 5CONTINUITY AND
DIFFERENTIABILITY
CONTINUITY AND DIFFERENTIABILITY 875.1.3Geometrical meaning of continuity (i) Functionf will be continuous atx =c if there is no break in the graph of the function at the point( ), ( )c f c. (ii) In an interval, function is said to be continuous if there is no break in the graph of the function in the entire interval.5.1.4Discontinuity
The functionf will be discontinuous atx =a in any of the following cases : (i)lim x a-® f (x) andlim x a+® f (x) exist but are not equal. (ii)lim x a-® f (x) andlim x a+® f (x) exist and are equal but not equal to f (a). (iii) f (a) is not defined.5.1.5Continuity of some of the common functions
Function f (x)Interval in which
f is continuous1. The constant function, i.e. f (x) =c
2. The identity function, i.e. f (x) =xR
3. The polynomial function, i.e.
f (x)= a0xn +a1xn-1 + ... +an-1x +an4. |x -a |(-¥,¥)
5.x-n,n is a positive integer(-¥,¥) - {0}
6.p (x) /q (x), wherep (x) andq (x) areR - {x :q (x) = 0}
polynomials inx7. sinx, cosxR
8. tanx, secxR- { (2n + 1)π
2:nÎZ}
9. cotx, cosecxR- { (np :nÎZ}
88 MATHEMATICS10.exR
11. logx(0,¥)
12. The inversetrigonometric functions,In their respective
i.e., sin -1x, cos-1x etc.domains5.1.6Continuity of composite functions
Letf andg be real valued functions such that (fog) is defined ata. Ifg is continuous ata andf is continuous atg(a), then (fog) is continuous ata.5.1.7Differentiability
The function defined byf¢(x) =0( ) ( )limhf x h f x h®+ -, wherever the limit exists, is
defined to be the derivative off atx. In other words, we say that a functionf is differentiable at a pointc in its domain if both0( ) ( )lim hf c h f c h -®+ -, called left hand derivative, denoted by L f¢ (c), and0( ) ( )lim hf c h f c h +®+ -, called right hand derivative, denoted by Rf¢ (c), are finite and equal. (i) The functiony =f (x) is said to be differentiable in an open interval (a, b) if it is differentiable at every point of (a, b) (ii) The functiony =f (x) is said to be differentiable in the closed interval [a, b] if Rf¢(a) and Lf¢ (b) exist andf¢ (x) exists for every point of (a,b). (iii) Every differentiable function is continuous, but the converse is not true5.1.8Algebra of derivatives
Ifu,v are functions ofx, then
(i)( )d u v d x±= ±du dv dx dx(ii)( )= +d dv duuv u vdx dx dx(iii)2du dv v ud udx dx dx v v-CONTINUITY AND DIFFERENTIABILITY 895.1.9 Chain rule is a rule to differentiate composition of functions. Letf =vou. If
t =u (x) and bothdt dx anddv dt exist then.=df dv dt dx dt dx5.1.10 Following are some of the standard derivatives (in appropriate domains) 1.-121(sin )1=-dxdxx2.-1
21(cos )1dxdxx-=-3.-1
21(tan )1=+dxdxx4.-1
21(cot )1dxdxx-=+5.-1
21(sec ), 11dxxdxx x=>-6.-1
21(cosec ), 11dxxdxx x-=>-5.1.11Exponential and logarithmic functions
(i) The exponential function with positive baseb> 1 is the function y =f (x) =bx. Its domain isR, the set of all real numbers and range is the set of all positive real numbers. Exponential function with base 10 is called the common exponential function and with basee is called the natural exponential function. (ii) Letb > 1 be a real number. Then we say logarithm ofa to baseb isx ifbx=a, Logarithm ofa to the baseb is denoted by logba. If the baseb = 10, we say it is common logarithm and ifb = e, then we say it is natural logarithms. logx denotes the logarithm function to basee. The domain of logarithm function isR+, the set of all positive real numbers and the range is the set of all real numbers. (iii) The properties of logarithmic function to any baseb> 1 are listed below:1. log
b (xy) = logbx + logby2. log
y = logbx - logby90 MATHEMATICS3. logbxn = nlogb x
4.logloglogc
b cx xb= , wherec > 15. log
bx1 log= xb6. logbb =1 and logb 1 = 0 (iv) The derivative ofexw.r.t.,x isex , i.e.( )x xde edx=. The derivative of logx w.r.t.,x is1 x; i.e.1(log )dxdx x=.5.1.12Logarithmic differentiation is a powerful technique to differentiate functions
of the formf (x) = (u (x))v(x), where bothf andu need to be positive functions for this technique to make sense.5.1.13Differentiation of a function with respect to another function
Letu =f (x) andv =g (x) be two functions ofx, then to find derivative of f (x) w.r.t. tog (x), i.e., to finddu dv, we use the formuladu dudx dvdv dx=.5.1.14Second order derivative2
2d dy d y
dx dx y2 , ify =f (x).
5.1.15Rolle"s Theorem
Letf : [a,b]®R be continuous on [a,b] and differentiable on (a,b), such thatf(a) =f (b), wherea andb are some real numbers. Then there exists at least one pointc in (a,b) such thatf¢ (c) = 0.CONTINUITY AND DIFFERENTIABILITY 91Geometrically Rolle"s theorem ensures that there is at least one point on the curve
y =f (x) at which tangent is parallel tox-axis (abscissa of the point lying in (a,b)).