MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction Te range is te set o all possible values o () as varies over te domain It is a subset o te domain A vertical line sould not cut te grap o a unction more tan once Composite Functions ( g o ) = g( ) or all in te domain o suc tat () is in te domain o g Hperbolic Functions e cos = + e Page 1 o 9
e sin = e Identities cos sin = 1 sin ( + = sin cos + cos sin cos ( + = sin sin + cos cos Derivatives d cos = sin d d sin = cos d Injective Functions A unction is injective (or one-to-one) i distinct elements in its domain are mapped to distinct elements in its codomain Tests 1 ( 1 ) ( ) or ( 1 ) = ( ) 1 = Te orizontal line test works i te unction is rom R to R Surjective Functions A unction is surjective (or onto) i its range is equal to its codomain Page o 9
Bijective Functions A unction is bijective i it is bot injective and surjective Inverse Functions Two unctions A B 1 : : and B A 1 are inverse unctions i ( o ) = or all 1 A and ( o ) = or all B A unction as an inverse i it is bijective Capter 4: Curves and Suraces in 3-dimensional Space Cartesian Coordinates in 3 Dimensions z Curves in 3-Dimensional Space A single equation tat relates, and z will, in general, represents a surace More tan one equation is needed to represent a curve in tree dimensions Tis is done b parametric equations Planes Te Cartesian equation o a plane in 3-dimensional space is o te orm a + b + c = d, were not all o a, b and c are zero Functions o Two Real Variables Te domain is a region in R and te codomain is R Te range is te set o all possible values taken b (, as (, varies over domain Page 3 o 9
Capter 5: Limits and te Limit Laws Deinition o a Limit Intuitive Deinition Suppose tat l is a real number Te limit o () as approaces c is equal to l, or lim = l (but not equal to c), i we can make () as close as we like to l or all suicientl close to c Matematicall Precise Deinition Suppose tat l is a real number Ten te limit o (), as approaces c, is equal to l i or an number ε > 0 we can alwas ind a number δ > 0 suc tat () l < ε wenever 0 < c < δ We irst coose ε and ten ind a δ so tat () l < ε wenever 0 < c < δ Ten tat limit olds true i or ever possible ε > 0, we can alwas ind suc a δ Te Squeeze Law A limit law: Suppose tat g, or all near c Ten lim g Te squeeze law: Suppose tat g lim = l and lim = l Ten g = l lim lim, or all near c, and tat Limits o Functions o Two Variables Intuitive Deinition ( ) l lim = i we can make (, as close as we like to l or all (, suicientl ( c, d ) close to, but not equal to, (c, d) Matematicall Precise Deinition Suppose tat l is a real number Ten, te limit o (,, as (, approaces (c, d), is equal to l i or eac ε > 0 we can ind a number δ > 0 suc tat (, l < ε wenever 0 < ( c) + ( d) < δ Te limit sould eist and be te same no matter wic direction ou come rom Limits at Ininit Suppose tat l is a real number Ten, te limit o (), as approaces, is equal to l i or all ε > 0 tere eists an N > 0 suc tat () l < ε wenever > N Te limit o (), as approaces, is equal to l i or all ε > 0 tere eists an N > 0 suc tat () l < ε wenever < N Page 4 o 9
Ininite Limits Te limit o (), as approaces c, is equal to i or ever positive number N > 0, tere eists a δ > 0 suc tat () > N wenever 0 < c < δ Te limit o (), as approaces c, is equal to i or ever positive number N > 0, tere eists a δ > 0 suc tat () < N wenever 0 < c < δ Te limit laws do not appl to ininite limits Capter 6: Continuous Functions Introduction Continuous unctions do not make sudden jumps or canges I is a continuous unction and i c belongs to te domain o ten = ( c) lim Continuit at a Point Suppose tat () is a unction and tat c is a point in te domain o () Ten () is continuous at = c i te limit lim and is equal to (c) I () is continuous at = c, ten (c) is inite Was in wic a unction can ail to be continuous at a point Te unction as a ole at te point It jumps Tere is an asmptote Continuous Functions A unction is continuous i it is continuous at ever point in its domain Tat is, () is continuous i = ( c) lim or all c in te domain o A unction is continuous everwere i it is continuous or all real c Composition Law Suppose tat () is a continuous unction and tat lim g = l ( ) = ( l) ( g) = lim g lim o Ten Page 5 o 9
L Hopital s Rule Suppose tat lim = 0, lim g = 0 inite Ten lim = lim c g c g Suppose tat lim = ±, g = ± inite Ten and tat te limit lim and tat te limit lim = lim c g g g lim eists and is c g lim eists and is c Te Intermediate Value Teorem Suppose tat () is continuous on te closed interval [a, b] Ten () takes ever value between (a) and (b) as varies between a and b Alternativel, i k is an number in between (a) and (b) ten () = k as at least one solution in te interval [a, b] Te IVT can be applied to approimate te solution o an equation Suppose tat is continuous on te interval [a, b] and tat (a) and (b) ave dierent signs Ten (c) = 0, or some c c [a, b] Capter 7: Dierentiable Functions Te Derivative o a Function A unction is dierentiable at = c i () c Te derivative o a unction is Dierentiable Functions and Continuit Ever dierentiable unction is continuous = ( c + ) ( c) lim eists 0 ( + ) = lim 0 Functions can ail to be dierentiable i Te unction is not dierentiable at a point in te given interval, and as a sarp point or a cusp Te unction canges too rapidl near a particular point Derivative o an Inverse Function Suppose tat is a dierentiable unction Ten -1 is dierentiable and d d 1 = 1 1 ( ) Page 6 o 9
Te Etreme Value Teorem Suppose tat is continuous on te closed interval [a, b] Ten as bot a minimum and a maimum value on [a, b]; tat is, tere eist numbers m and M suc tat m () M, or all c [a, b], m = (c) and M = (d) or some c, d c [a, b] Te range o on [a, b] is [m, M] A unction as a local minimum at = c i tere eists a δ > 0 suc tat () (c) or all c (c δ, c + δ) Te same goes or local maimum Suppose tat is dierentiable on [a, b] and tat c c [a, b] is a maimum or a minimum value or on [a, b] Ten eiter c = a, c = b or (c) = 0 Te Mean Value Teorem Suppose tat is dierentiable on [a, b] Ten tere eists a number c c (a, b) suc ( b) ( a) tat () c = b a Capter 8: Talor Polnomials Constructing Talor Polnomials about = 0 Te nt degree Talor polnomial o a unction around = 0 is T n ( 0) + ( 0) ( n ( 0) ( 0) ) ( 0) n 3 = + + + +! 3! n! Te nt degree Talor polnomial o a unction around = a is T n = ( a) + ( a) Te Remainder Term +! ( n ( a) ( ) ( ) ) a ( ) 3 ( a) ( ) n a + 3! a Te dierence between and T n is R T Te Lagrange orm o te remainder is n = n ( n+ 1 ) ( c) ( n + 1 )! + + n+ 1 R n = n! a Capter 10: Partial Derivatives and Tangent Planes Let be a unction o two variables Ten te partial derivative o wit respect to is = = lim 0 ( +, Let be a unction o two variables Ten te partial derivative o wit respect to is = = lim 0 + ) Page 7 o 9
I (a, b) belongs to te domain o, and eist and are continuous wen = a and = b ten is dierentiable at (a, b) Higer Order Partial Derivatives ( ) = = ( ) = ( ) = = = ( ) = = Tangent Planes Te equation o te tangent plane is z ( a b) = ( a, b)( a) + ( a, b)( b), Linear Approimation Using Dierentials Te dierential o a unction o two variables is given b dz = d + d z z Te Total Derivative Rule Suppose tat z = (, were = g(t) and = (t) and, g and are all dierentiable unctions Ten te total derivative o z is given b dz dt z d z d + dt dt = Implicit Dierentiation d d = Capter 1: Directional Derivatives and te Gradient Vector Te Gradient Vector and te Directional Derivative grad Te gradient o is (, ) = (, ) = i + j Te directional derivative o in te direction o a unit vector u is D u = u = u1 + u ( a + u, b + u ) = lim 0 1 Page 8 o 9
Interpreting te Gradient Vector D u ( a, b) = ( a, b) cosθ wereθ is te angle between and u Hence te greatest value o te directional derivative is ( a, b) Te direction o steepest increase o (, is It will take its largest negative value wen u is in te opposite direction to Te tangent wit zero slope lies in te direction at rigt angles to te gradient vector is normal to te level curve (, = c Page 9 o 9