UNIVERSIDAD CARLOS III DE MADRID Escuela Politécnica Superior Departamento de Matemáticas

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1 UNIVERSIDAD CARLOS III DE MADRID Escuela Politécnica Superior Departamento de Matemáticas a t e a t i c a s PROBLEMS, CALCULUS I, st COURSE. FUNCTIONS OF A REAL VARIABLE BACHELOR IN: Audiovisual System Engineering Communication System Engineering Telematics Engineering Set prepared by Arturo de PABLO Elena ROMERA English version by Marina DELGADO Javier RODRÍGUEZ

2 Functions of a real variable Functions of a real variable. The real line Problem.. i) Consider the real numbers 0 < a < b, k > 0. Prove the inequalities ) a < ab < a + b < b, ) a b < a + k b + k. ii) Prove that a + b = a + b ab 0. Prove the inequality a b a b, for all a, b IR. iv) Prove that: a) ma{, y} = + y + y, b) min{, y} = + y y. v) Epress in a single formula the following function f() = () + = { if > 0 0 if 0. Problem.. n IN we have Decompose the epressions in n as a product of factors to prove that, for all i) n n is even; ii) n 3 n is a multiple of 6; n is a multiple of 6 if n is even. Problem..3 Use the induction technique to prove the following formulas: i) n j = j= n(n + ) ; ii) n (j ) = n ; j= n j = j= n(n + )(n + ). 6 Problem..4 i) Geometrical sum: Prove by induction n j=0 r j = rn+, for all n IN, r ; r ii) Bernouilli inequality: ( + h) n + nh, for all n IN, h >. Problem..5 Prove by induction that for all n IN we have i) 0 n is a multiple of 9; ii) 0 n ( ) n is a multiple of. Problem..6 i) Prove that a number is a multiple of 4 if and only if its two last figures make up a multiple of 4. ii) Prove that a number is a multiple of k if and only if its k last figures make up a multiple of k.

3 Functions of a real variable Prove that a number is a multiple of 3 (or 9) if and only if the sum of its figures is a N multiple of 3 (or 9). In other terms, n = a j 0 j is a multiple of 3 (or 9) if and only if N a j is a multiple of 3 (or 9). j=0 j=0 iv) Prove that a number is a multiple of if and only if the sum of its even placed figures N minus the sum of its odd placed figures is a multiple of, i.e.: ( ) j a j is a multiple of. Hints: ii) write the number in the form n = 0 k a + b, with a 0, 0 b < 0 k ; and iv) use problem..5. j=0 Problem..7 that Prove by induction and using Newton s binomial that for all n IN we have i) n 3 n is a multiple of 6; ii) n 5 n is a multiple of 5. Problem..8 Prove that: i) if n IN is not a perfect square, n / Q; ii) + 3 / Q. Hint: i) write n = z r, where r does not contain any square factor. Problem..9 Find the set of IR that verify: i) A = { 3 8 }, ii) B = { 0 < < / }, C = { }, iv) D = { 3 ( + 3)( 5) < 0 }, v) E = { > 0 }, vi) F = { 4 < }, vii) G = { 4 < }, v H = { < }, i) I = { + = 0 }, ) J = { + > }. Problem..0 Given two real numbers a < b, let us define, for each t IR the number (t) = ( t)a + tb. Describe the following sets of numbers: i) A = { (t) : t = 0,, / }, ii) B = { (t) : t (0, ) }, C = { (t) : t < 0 }, iv) D = { (t) : t > }.

4 Functions of a real variable 3 Problem.. Find the supremum, the infimum, the maimum and the minimum (if they eist) of the following sets of real numbers: i) A = { } [, 3); ii) B = {3} {} { } [0, ]; C = { = + /n : n IN }; iv) D = { = (n + )/n : n IN }; v) E = { IR : }; vi) F = { IR : ( a)( b)( c)( d) < 0 }, vii) G = { = p + 5 q : p, q IN }; v H = { = ( ) n + /m : n, m IN }. with a < b < c < d fied; Problem.. Represent in IR the following sets: i) A = { y < }, ii) B = { < y < }, C = { + y ZZ }, iv) D = { + y = }, v) E = { ( ) + (y + ) < 4 }, vi) F = { = y }, vii) G = { 4 + y 4, y 0 }, v H = { + y < 9, y 0 }. Problem..3 mn =. Problem..4 with a, b, c > 0. Prove that the straight lines y = m + b, y = n + c are orthogonal if Consider the plane triangle defined by the points (a, 0), ( b, 0) and (0, c), i) Compute the intersection point of the three altitudes. ii) Calculate the intersection point of the three medians. When do these two points coincide? Problem..5 i) Consider the parabola G = {y = } and the point P = (0, /4). Find λ IR such that the points of G are equidistant from P and the horizontal line L = {y = λ}. ii) Conversely, find that the set G such that its points are equidistant from a point P = (a, b) and a straight line L = {y = λ}, is the parabola y = α + β + γ. Find α, β, γ. Problem..6 i) Find the set of points in the plane such that the sum of their distances to the points F = (c, 0) and F = ( c, 0) is a, (a > c). ii) Same question, substituting sum by difference (with a < c).

5 Functions of a real variable 4. Elementary functions Problem.. Problem.. Find the domain of the following functions: i) f() = 5 + 6, ii) f() = +, f() =, iv) f() = v) f() = 4, log, vi) f() = log( ), 5 vii) f() =, v f() = arcsin(log ). log i) If both f and g are odd functions, what are f + g, f g and f g? ii) What if f is even and g is odd? Problem..3 Study the symmetry of the following functions: i) f() =, ii) f() = +, Hint: vi) is odd. f() = sin, iv) f() = (cos 3 )(sin )e 4, v) f() = +, vi) f() = log( + ). Problem..4 For which numbers a, b, c, d IR does the function f() = a + b c + d satisfy f f = I (identity) on the domain of f? Problem..5 Check that the function f() = + 3 is bijective, defined from IR { /} + to IR {/} and find its inverse. Problem..6 i) Study which of the following functions are injective, finding their inverses when they have them, or give an eample of two points with the same image otherwise. a) f() = 7 4, b) f() = sin(7 4), c) f() = ( + ) 3 +, d) f() = + +, e) f() = 3 +, f) f() = +, g) f() = e, h) f() = log( + ).

6 Functions of a real variable 5 ii) Prove that the function f() = 3 + is injective on (3/, ). Prove that the function f() = + is injective on (, ) and find f ( /3). iv) Study whether the previous functions are surjective and bijective on their domain D(f) in IR. Problem..7 Prove that a sin + b cos can be written as A sin( + B), and find A and B. Problem..8 Calculate i) arctg + arctg 3, ii) arctg + arctg 3, arctg + arctg 5 + arctg 8. Hint: use the formula for the tangent of a sum and study the signs. Problem..9 Simplify the following epressions i) f() = sin(arccos ), ii) f() = sin( arcsin ), f() = tg(arccos ), iv) f() = sin( arctg ), v) f() = cos( arctg ), vi) f() = e 4 log. Problem..0 Solve the following system of equations, for, y > 0, { y = y y = 3. Problem.. Describe function g in terms of f in the following cases (c IR is a constant). Plot them for f() = and f() = sin. i) g() = f() + c, ii) g() = f( + c), g() = f(c), iv) g() = f(/), v) g() = f( ), vi) g() = f(), vii) g() = /f(), v g() = (f()) + = ma{f(), 0}. Problem.. Sketch, with as few calculations as possible, the graph of the following func-

7 Functions of a real variable 6 tions: i) f() = ( + ), ii) f() = 4, f() = + /, iv) f() = /( + ), v) f() = min{, }, vi) f() = e, vii) f() = [], v f() = /[/], i) f() =, ) f() = e, i) f() = log( ), ii) f() = sin(/). Hint: [] = n denotes the integer part of, i.e., the biggest integer n. Problem..3 Let us define the hyperbolic functions sinh = e e, cosh = e + e. i) Study their symmetry. ii) Prove the formulas a) cosh sinh =, b) sinh = sinh cosh. Simplify the function f() = sinh. iv) Sketch the graph of the functions sinh and cosh. Problem..4 Sketch the following curves given in polar coordinates: i) r =, θ [0, π], ii) θ = 3π/4, r, r = sin θ, θ [0, π], iv) r = θ, θ [0, π], v) r = e θ, θ [ π, π], vi) r = sec θ, θ [0, π/], vii) r = sin θ, θ [0, π], v r = (cos θ) +, θ [0, π], i) r = cos θ, θ [0, π], ) r = (sin 3θ) +, θ [0, π/3]. Problem..5 Sketch the following subsets of the plane given in polar coordinates: i) A = { < r < 4 }, ii) B = { π/6 θ π/3 }, C = { r θ, 0 θ 3π/ }, iv) D = { r sec θ, 0 θ π/4 }.

8 Functions of a real variable 7.3 Limits of functions Problem.3. Using the ε δ definition of it, prove that: i) = 4, ii) 3 (5 ) 6, + sin = 0, iv) = 3. 9 Problem.3. Find the following its simplifying the common factors which might appear: n a n i) a a, ii) a a a, , iv) 4, v) h 0 ( h) 3 h (, vi) ). Problem.3.3 Using the its sin =, e =, find the following its: i) (sin 3 ) 6, ii) cos, tg + sin( + a) sin a +, iv), log( + ) v), vi) ( + ) /, vii) i) i) log( ), v ( + sin ) /, sin e e sin tg sin, ) sin 3, ( ) sin sin, ii) (cos ) /, sin sin(/) a b π ( π), iv). Hint: it may be necessary to use a change of variables and the it of the composite function.

9 Functions of a real variable 8 Problem.3.4 Calculate the following its: i) , ii) sin , + ( +, iv) + 4 ), v) e e, vi) e e, vii), v Problem.3.5 Problem.3.6 Problem.3.7 Find the one-sided its: ( ) [t], ( i) ii) t 0 + t t 0 t Find the its t 0 e/t, iv) + t 0 e/t. ) [t], ( + 7 ) 4 i) + 3 e /t, ii) 6 t 0 + e /t i) Establish the relation between a and b so that a/( ) = (cos ) b/. ii) If f() = log(log ) and α > 0, find (f() f(α)) and (f() f(α )). Problem.3.8 i) Prove that if f() = 0 then f() sin / = 0. ii) Calculate + sin /..4 Continuity Problem.4. i) Prove that if f is continuous at a point a and g is at f(a), then g f is continuous at a. ii) Prove that if f is continuous, then f is also. Is the reciprocal true? What can be said of a function that only takes values on Q? Problem.4. Find λ IR so that the function b() = i) IR, ii) [0, ]. λ λ + is continuous on:

10 Functions of a real variable 9 Problem.4.3 Study the continuity of the following functions: i) f() = e 5 + cos 8 + ; ii) g() = e 3/ + 3 9; h() = 3 tg(3 + ); iv) j() = 5 + 6; v) k() = (arcsin ) 3 ; vi) m() = ( 5) log(8 3); Problem.4.4 Study the continuity of the following functions: i) f() = []; { ii) f() = sin(/) if 0 0 if = 0; tg if > 0 f() = 0 if = 0 e / if < 0; { if Q iv) f() = if / Q. Problem.4.5 Prove the following fied points theorems: i) Let f : [0, ] [0, ] a continuous function. Then there eists c [0, ] such that f(c) = c. ii) Let f, g : [a, b] IR two continuous functions such that f(a) > g(a), f(b) < g(b). Then there eists c (a, b) such that f(c) = g(c).