Maths WorkshopMaths problems with full solutions

Calculus problems

504 problems · Limits, continuity, derivatives, integrals, series and differential equations, in one and several variables.

Ex.1A functional equation with f(2x)
#functional equations #continuity at a point #sequences
Functional equations
L1
L1
Ex.2A recurrence and a proof by induction
#induction #recurrence #closed form
Sequences
High school
High school
Ex.3Cauchy's equation f(x + y) = f(x) + f(y)
#functional equations #density #rational numbers
Functional equations
L1
L1
Ex.4Continuity with parameters
#continuity #piecewise functions #parameters
Continuous and discontinuous functions
High school
High school
Ex.5When do integrals of powers converge?
#improper integrals #p-test #Riemann integrals
Improper integrals
L1
L1
Ex.6Extending a rational function by continuity
#extension by continuity #factoring #removable singularity
Continuous and discontinuous functions
High school
High school
Ex.7An integral whose integrand blows up
#improper integrals #logarithm #limits
Improper integrals
L1
L1
Ex.8Polynomial bounds for sine and cosine
#inequalities #monotonicity
Differentiable functions
L1
L1
Ex.9Terms that do not tend to zero
#divergence test #necessary condition
Convergence tests
L1
L1
Ex.10Expanding √x at x = 4
#Taylor expansion #shifted point #numerical estimate
Taylor expansions
L1
L1
Ex.15A fence along a river
#optimisation #area #quadratic function
Optimisation and curve sketching
High school
High school
Ex.16f(x)f(y) − f(xy) = x + y
#substitution #special values #verification
Functional equations
L1
L1
Ex.17The mean value theorem
#mean value theorem #Lagrange #existence
Mean Value Theorem
High school
High school
Ex.18Rolle's theorem on a polynomial
#Rolle #polynomial #existence
Mean Value Theorem
High school
High school
Ex.19Step functions and their integrals
#step function #partition #integral
Riemann integral
High school
High school
Ex.20Integral of a constant step function
#constant #step function #elementary
Riemann integral
High school
High school
Ex.21An improper integral with exponentials
#improper integral #exponential #partial fraction
Improper integrals
L1
L1
Ex.22Three basic limits by conjugation and standard limits
#conjugate #standard limit #sine
Limits
High school
High school
Ex.24Equivalents, and a sign trap
#equivalents #comparison test #positive series
Convergence tests
L1
L1
Ex.25A fixed point for every continuous self-map of [0,1]
#fixed points #intermediate value theorem
Intermediate Value Theorem
High school
High school
Ex.26Extensions by continuity and oscillation
#extension by continuity #oscillation #squeeze theorem
Continuous and discontinuous functions
L1
L1
Ex.27Locating the roots of a cubic
#intermediate value theorem #polynomial roots #bracketing
Intermediate Value Theorem
High school
High school
Ex.28How long is the drug effective?
#applications #intermediate value theorem #bracketing
Intermediate Value Theorem
High school
High school
Ex.29An improper integral of a rational function
#improper integrals #arctangent #convergence
Improper integrals
M1
M1
Ex.30Where is 1/ln|x| continuous?
#continuity #limits #logarithm
Continuous and discontinuous functions
L1
L1
Ex.31Which of these functions extend by continuity?
#extension by continuity #one-sided limits #exponential
Continuous and discontinuous functions
L1
L1
Ex.32lim (1/x² − 1/sin² x)
#limits #expansions #difference of fractions
Taylor expansions
L1
L1
Ex.33lim ((1 + x)^(1/x) / e)^(1/x)
#limits #indeterminate form 1^∞ #logarithm #expansions
Taylor expansions
L2
L2
Ex.34Summary of continuity
#summary #continuity
Continuity
L2
L2
Ex.42lim cos(1/n)^(n²)
#sequences #expansions #indeterminate form 1^∞
Taylor expansions
L1
L1
Ex.43Mean value theorem for $f(x)=x^3$
#mean value theorem #cubic #existence
Mean Value Theorem
High school
High school
Ex.44Summary of the mean value theorem
#summary #MVT #Rolle
Mean Value Theorem
High school
High school
Ex.45Additivity over adjacent intervals
#additivity #step function #split
Riemann integral
High school
High school
Ex.46A step function built from the floor
#step function #floor function #integral
Riemann integral
High school
High school
Ex.47Linearity of the integral on step functions
#linearity #step functions #sum
Riemann integral
High school
High school
Ex.48Monotonicity of the integral
#monotonicity #step functions #comparison
Riemann integral
High school
High school
Ex.49Summary of step functions and Riemann integration
#summary #step functions #Riemann integrability
Riemann integral
High school
High school
Ex.50Two subdivisions for the same function
#step function #subdivision #independence
Riemann integral
High school
High school
Ex.51Four limits with absolute value and trigonometry
#absolute value #one-sided limits #factorization
Limits
High school
High school
Ex.52Three more classical limits
#squeeze theorem #logarithm #exponential form
Limits
L1
L1
Ex.53Proving convergence by comparison
#improper integrals #comparison test #divergence
Improper integrals
L1
L1
Ex.54Floor plus square root of the fractional part
#continuity #floor function #one-sided limits
Continuous and discontinuous functions
L1
L1
Ex.55The monk on the mountain
#intermediate value theorem #classic puzzle
Intermediate Value Theorem
High school
High school
Ex.56Expansions of 1/(1 + x + x²) and ln(1 + x + x²)
#Taylor expansion #geometric series #factorisation trick
Taylor expansions
L1
L1
Ex.62Position of a curve relative to its tangent
#tangent line #inflection #local behaviour
Taylor expansions
L1
L1
Ex.63Integral of $|f|$ for a step function
#absolute value #step function #triangle inequality
Riemann integral
High school
High school
Ex.64Darboux sums for a step function
#Darboux sums #upper lower #step functions
Riemann integral
L1
L1
Ex.65A step function with zero integral
#zero integral #step functions #nonnegative
Riemann integral
High school
High school
Ex.66Squeezing between step functions
#Riemann integrability #step functions #squeeze
Riemann integral
L1
L1
Ex.67A convergent improper integral of $\ln x/(1+x)^2$
#improper integral #integration by parts #logarithm
Improper integrals
L1
L1
Ex.68An improper integral of $\ln x/x^2$ by parts
#improper integral #integration by parts #logarithm
Improper integrals
L1
L1
Ex.69Three limits involving logarithm and exponentials
#exponential form #logarithm #indeterminate forms
Limits
L1
L1
Ex.70The midpoint functional equation
#functional equations #affine functions
Functional equations
L1
L1
Ex.75Composition with a continuous function
#composition #continuous #integrability preservation
Riemann integral
L1
L1
Ex.76A monotone function on a fine grid
#monotone functions #Riemann integrability #step approximation
Riemann integral
L1
L1
Ex.77Continuity of a rational function
#rational function #domain #continuity
Continuity
High school
High school
Ex.78Glued along the rationals and the irrationals
#continuity #density #sequences #pathological functions
Continuous and discontinuous functions
L1
L1
Ex.79The limit of an iterated sequence is a fixed point
#sequences #fixed points #continuity #golden ratio
Continuous and discontinuous functions
L1
L1
Ex.80Maximum and minimum of two continuous functions
#continuity #absolute value #algebra of continuous functions
Continuous and discontinuous functions
L1
L1
Ex.81The absolute value of a continuous function
#continuity #absolute value #counterexamples
Continuous and discontinuous functions
L1
L1
Ex.82The ε–N definition of a limit
#definition of a limit #epsilon-N #divergence
Sequences
L1
L1
Ex.83How many terms are needed to compute sin 1?
#Taylor-Lagrange #error bound #numerical computation
Taylor expansions
L1
L1
Ex.86How many trees in the orchard?
#optimisation #modelling #quadratic function
Optimisation and curve sketching
High school
High school
Ex.87Fixed point on [0,1]
#fixed point #IVT #auxiliary function
Intermediate Value Theorem
High school
High school
Ex.88A continuous periodic function is bounded
#extreme value theorem #periodic functions #boundedness
Continuous and discontinuous functions
L1
L1
Ex.93Additive and multiplicative functions
#Cauchy equation #squares are nonnegative #monotonicity
Functional equations
L1
L1
Ex.94Cauchy's equation with monotonicity
#Cauchy equation #rationals #density
Functional equations
Olympiad
Olympiad
Ex.96A parameter integral equal to arctan
#parameter integral #fundamental theorem #arctan
Riemann integral
L1
L1
Ex.97A limit via the mean value theorem for integrals
#mean value theorem for integrals #logarithm #limit
Riemann integral
L1
L1
Ex.100f(x + y) + f(x − y) = 2f(x)f(y)
#d'Alembert #cosine #differential equation #differentiability
Functional equations
Olympiad
Olympiad
Ex.101f(x² + f(y)) = y + f(x)²
#injectivity #surjectivity #oddness #Cauchy equation
Functional equations
Olympiad
Olympiad
Ex.102When the equivalence rule for improper integrals fails
#improper integral #integration by parts #absolute convergence #conditional convergence
Improper integrals
L1
L1
Ex.103Limit of an integral with arctan
#mean value theorem for integrals #arctan #limit
Riemann integral
L1
L1
Ex.104Limit of an integral with cosine and logarithm
#mean value theorem for integrals #cosine #logarithm
Riemann integral
L1
L1
Ex.105Commuting maps on $[0,1]$ share a fixed point
#commuting maps #IVT #extreme value theorem #induction
Intermediate Value Theorem
L1
L1
Ex.106Lipschitz functions are uniformly continuous
#Lipschitz #uniform continuity #epsilon-delta
Continuous and discontinuous functions
L1
L1
Ex.107Coordinate projections are continuous
#projection #continuity #coordinates
Continuity
L1
L1
Ex.109Distance to a line, without a formula
#optimisation #distance #line #quadratic function
Optimisation and curve sketching
High school
High school
Ex.110Continuous functions with f^2=1 on an interval
#IVT #sign #constant functions
Intermediate Value Theorem
High school
High school
Ex.1111/x on (0,1] is not uniformly continuous
#uniform continuity #open intervals #Heine theorem
Continuous and discontinuous functions
L1
L1
Ex.112The square root: uniformly continuous, not Lipschitz
#uniform continuity #Lipschitz #square root
Continuous and discontinuous functions
L1
L1
Ex.113x² is not uniformly continuous on the real line
#uniform continuity #sequences #counterexamples
Continuous and discontinuous functions
L1
L1
Ex.114A continuous f with f o f= cos cannot exist
#functional equation #cosine #continuity #monotonicity
Functional equations
L1
L1
Ex.116Least squares for a line through the origin
#least squares #regression #statistics #quadratic
Optimisation and curve sketching
L1
L1
Ex.117Limits of sums as Riemann sums
#Riemann sums #limits #pi #ln 2
Riemann integral
L1
L1
Ex.119A limit of a product via Riemann sums
#Riemann sums #logarithm #limit #product
Riemann integral
L1
L1
Ex.121The most economical speed of a truck
#optimisation #modelling #economics #cost per kilometre
Optimisation and curve sketching
High school
High school
Ex.123The economic order quantity
#optimisation #economics #inventory #modelling
Optimisation and curve sketching
L1
L1
Ex.125Maximising profit
#optimisation #economics #marginal cost #modelling
Optimisation and curve sketching
L1
L1
Ex.127The strongest beam cut from a log
#optimisation #geometry #engineering #Pythagoras
Optimisation and curve sketching
L1
L1
Ex.129A window with a semicircular top
#optimisation #area #perimeter #geometry
Optimisation and curve sketching
L1
L1
Ex.132The trapezoid of largest area with three equal sides
#optimisation #trigonometry #geometry #trapezoid
Optimisation and curve sketching
L1
L1
Ex.135The triangle of least area through a point
#optimisation #lines #triangle #area
Optimisation and curve sketching
L1
L1
Ex.137Among triangles with a given base and perimeter, the isosceles one is the largest
#optimisation #Heron's formula #triangle
Optimisation and curve sketching
L2
L2
Ex.139The largest rectangle inscribed in an ellipse
#optimisation #ellipse #parametrisation #trigonometry
Optimisation and curve sketching
L1
L1
Ex.141The largest cone and the largest cylinder inscribed in a sphere
#optimisation #volume #sphere #solids
Optimisation and curve sketching
L1
L1
Ex.142The paper cup of largest capacity
#optimisation #cone #volume #geometry
Optimisation and curve sketching
L1
L1
Ex.144At which height should a lamp be hung?
#optimisation #physics #illumination #Lambert's law
Optimisation and curve sketching
L2
L2
Ex.147Maximum sustainable yield
#optimisation #modelling #ecology #logistic growth
Optimisation and curve sketching
L2
L2
Ex.149The shortest path touching a line
#optimisation #reflection #geometry #sums of square roots
Optimisation and curve sketching
L1
L1
Ex.152The closest point of an ellipse
#optimisation #ellipse #distance #constrained minimum
Optimisation and curve sketching
L1
L1
Ex.154Minimising a/x + b/(1 − x)
#optimisation #inequalities #Cauchy–Schwarz #parameters
Optimisation and curve sketching
L2
L2
Ex.156The best constant approximating x²
#minimax #uniform approximation #absolute value
Optimisation and curve sketching
L2
L2
Ex.158Firing up a slope: the best angle
#optimisation #trigonometry #physics #projectile
Optimisation and curve sketching
L2
L2
Ex.159The maximum of 3 sin x + 4 cos x
#extrema #trigonometry #amplitude
Optimisation and curve sketching
L1
L1
Ex.161Global extrema of x³ − 3x on [−2, 3]
#global extrema #closed interval #critical points
Optimisation and curve sketching
High school
High school
Ex.163Extrema of |x² − 3x + 2| on [0, 3]
#absolute value #global extrema #non-differentiable points
Optimisation and curve sketching
High school
High school
Ex.167The maximum of xⁿe^(−x)
#extrema #inequalities #exponential #growth
Optimisation and curve sketching
L1
L1
Ex.168The minimum of x^x
#logarithmic differentiation #extrema #limits
Optimisation and curve sketching
L1
L1
Ex.171A family of functions: x² + m/x
#parameters #extrema #monotonicity #discussion
Optimisation and curve sketching
L1
L1
Ex.172Complete study of x e^(−x)
#curve sketching #inflection point #number of solutions
Optimisation and curve sketching
L1
L1
Ex.175Complete study of (ln x)/x
#curve sketching #asymptotes #inflection point #logarithm
Optimisation and curve sketching
L1
L1
Ex.176Sketching x³/(x² − 1)
#curve sketching #oblique asymptote #odd function #inflection point
Optimisation and curve sketching
L2
L2
Ex.177Sketching x√(1 − x²)
#curve sketching #vertical tangent #odd function #domain
Optimisation and curve sketching
L2
L2
Ex.178How many solutions does eˣ = ax have?
#number of solutions #parameter #tangent #extrema
Optimisation and curve sketching
L1
L1
Ex.184The exponential lies above its tangent
#inequalities #derivatives
Differentiation
L1
L1
Ex.187Integral of sine squared
#integrals #trigonometry
Integration techniques
L1
L1
Ex.188Integration by parts
#integrals #by parts
Integration techniques
L1
L1
Ex.192(e^(2x) − 1) over x
#exponential #change of variable
Limits
L1
L1
Ex.193A limit by factoring
#indeterminate forms #factoring
Limits
High school
High school
Ex.194Limits of rational functions at infinity
#limits at infinity #rational functions
Limits
High school
High school
Ex.195One-sided limits of |x|/x
#one-sided limits #absolute value
Limits
High school
High school
Ex.196sin(3x) over 5x
#fundamental trigonometric limit #change of variable
Limits
High school
High school
Ex.197Sine over x at infinity
#squeeze theorem #limits at infinity
Limits
High school
High school
Ex.198A telescoping sum
#telescoping #series #partial sums
Sequences
High school
High school
Ex.199Area between a parabola and a line
#area #definite integrals #intersection points
Riemann integral
High school
High school
Ex.200An area with an absolute value
#definite integrals #absolute value #sign of a function
Riemann integral
High school
High school
Ex.201Arithmetic and geometric sequences
#arithmetic sequence #geometric sequence #sums
Sequences
High school
High school
Ex.202Recognising antiderivatives
#antiderivatives #definite integrals #fundamental theorem
Riemann integral
High school
High school
Ex.203Derivative of a quotient
#derivative #quotient rule
Differentiable functions
High school
High school
Ex.207Geometric Series
#geometric series #convergence #summation
Series
High school
High school
Ex.208Geometric series and repeating decimals
#geometric series #decimals #sum
Find the sum
High school
High school
Ex.209Geometric series with a parameter
#geometric series #convergence domain #parameters
Find the sum
High school
High school
Ex.210Integrals that simplify by parity
#symmetry #odd and even functions
Riemann integral
High school
High school
Ex.211Rolle's theorem on a quartic
#Rolle's theorem #polynomials
Mean Value Theorem
L1
L1
Ex.212The sum of the first n squares
#induction #sums #limits
Sequences
High school
High school
Ex.213Derivative and tangent line of a cubic
#derivative #tangent line #polynomials
Differentiable functions
High school
High school
Ex.214The derivative from the definition
#definition of the derivative #difference quotient #square root
Differentiable functions
High school
High school
Ex.225Continuity of a polynomial
#continuity #polynomial #continuity everywhere
Continuity
High school
High school
Ex.256Double integral over a rectangle
#double integral #rectangle #iterated
Multiple integrals
High school
High school
Ex.257Evaluating partial derivatives at a point
#partial derivatives #evaluation #point
Partial derivatives
High school
High school
Ex.258First partial derivatives
#partial derivatives #definition #computation
Partial derivatives
High school
High school
Ex.314A limit with sine
#limits #taylor
Limits
L2
L2
Ex.316A cubic with exactly one real root
#intermediate value theorem #monotonicity
Intermediate Value Theorem
L1
L1
Ex.318A limit with the exponential
#limits #taylor
Limits
L2
L2
Ex.321An improper integral
#improper integrals #integration by parts
Improper integrals
L2
L2
Ex.332Radius of convergence
#power series #ratio test
Power series
L2
L2
Ex.343The logarithm lies below its tangent
#inequalities #derivatives
Differentiation
L1
L1
Ex.344The conjugate trick at infinity
#conjugate #square roots #infinity minus infinity
Limits
L1
L1
Ex.345The conjugate trick at zero
#conjugate #square roots #indeterminate forms
Limits
High school
High school
Ex.346Epsilon-delta for a linear function
#epsilon-delta #definition of limit
Limits
L1
L1
Ex.347(e^x − cos x) over sin x
#combining known limits #indeterminate forms
Limits
L1
L1
Ex.348One minus cosine over x squared
#trigonometric limits #half-angle formula
Limits
L1
L1
Ex.349Sine is 1-Lipschitz
#mean value theorem #inequalities
Mean Value Theorem
L1
L1
Ex.350The n-th root of n
#sequences #logarithm #growth comparison
Sequences
L1
L1
Ex.351(2^x − 1) over x
#exponential #logarithm #derivative
Limits
L1
L1
Ex.352x ln(1 + 1/x) at infinity
#logarithm #number e #limits at infinity
Limits
L1
L1
Ex.353x times sin(1/x)
#squeeze theorem #oscillation
Limits
L1
L1
Ex.354A continuous function that vanishes on the rationals
#density #continuity #sequences
Continuous and discontinuous functions
L1
L1
Ex.355A linear recurrence of order two
#linear recurrence #characteristic equation
Sequences
L1
L1
Ex.356A telescoping product
#telescoping #products #factoring
Sequences
L1
L1
Ex.357A telescoping sum with factorials
#telescoping #factorials
Sequences
L1
L1
Ex.358A square root whose difference tends to 1/2
#divergence test #square roots #asymptotics
Convergence tests
M1
M1
Ex.359An arithmetico-geometric sequence
#recurrence #fixed point #geometric sequence
Sequences
L1
L1
Ex.360Odd degree polynomials have a real root
#intermediate value theorem #polynomials #limits at infinity
Intermediate Value Theorem
L1
L1
Ex.361The Bertrand series, all cases
#Bertrand series #parameters #integral test #comparison
Convergence tests
L1
L1
Ex.362Derivative of an inverse function
#inverse function #implicit differentiation #arcsine
Differentiable functions
L1
L1
Ex.363A surprisingly simple derivative
#chain rule #composition #square roots
Differentiable functions
L1
L1
Ex.364Derivative with an absolute value
#differentiability #absolute value #one-sided derivatives
Differentiable functions
L1
L1
Ex.365Differentiability when gluing two pieces
#differentiability #piecewise functions #one-sided derivatives
Differentiable functions
L1
L1
Ex.372Splitting the sum of 1/n² into even and odd terms
#absolute convergence #rearrangement #pi squared
Find the sum
L1
L1
Ex.373A separable first-order equation
#separable equation #integration #initial condition
First-order equations
L1
L1
Ex.374Integration by parts, two classics
#integration by parts #definite integrals
Integration techniques
L1
L1
Ex.375Three limits by the squeeze theorem
#squeeze theorem #sums #n-th roots
Sequences
L1
L1
Ex.376Logarithmic differentiation
#derivative #exponential form #powers with variable exponent
Differentiable functions
L1
L1
Ex.377The mean value theorem and an arctangent inequality
#mean value theorem #inequalities #arctangent
Mean Value Theorem
L2
L2
Ex.378Two-sided bounds for the logarithm
#mean value theorem #inequalities #logarithm
Mean Value Theorem
L1
L1
Ex.379Double integral over a triangular domain
#double integral #iterated integration #triangular domain
Multiple integrals
L1
L1
Ex.380Partial derivatives and the gradient
#partial derivatives #gradient #chain rule
Partial derivatives
L1
L1
Ex.381Simplifying and decomposing rational integrands
#partial fractions #rational functions
Integration techniques
L1
L1
Ex.382Complex conjugate roots
#characteristic equation #complex roots #oscillation
Linear equations of order two
L1
L1
Ex.383A double root
#characteristic equation #double root #homogeneous equation
Linear equations of order two
L1
L1
Ex.384Distinct real roots
#characteristic equation #real roots #homogeneous equation
Linear equations of order two
L1
L1
Ex.385Series with factorials and the number e
#exponential series #factorials #e
Find the sum
L1
L1
Ex.386Integration by substitution
#substitution #change of variable
Integration techniques
L1
L1
Ex.387Tangent lines through an external point
#tangent line #parabola #geometry
Differentiable functions
High school
High school
Ex.388Telescoping series with partial fractions
#telescoping #partial fractions #sums
Find the sum
L1
L1
Ex.389Telescoping with logarithms
#telescoping #logarithm #divergence
Find the sum
L1
L1
Ex.390Telescoping with a shift p
#telescoping #partial fractions #harmonic numbers
Find the sum
L1
L1
Ex.391Telescoping with square roots
#telescoping #square roots #divergence #p-series
Convergence tests
L1
L1
Ex.392The bisection method in practice
#bisection #numerical methods #intermediate value theorem
Intermediate Value Theorem
L1
L1
Ex.393The comparison test in practice
#comparison test #p-series
Convergence tests
L1
L1
Ex.394The derivative of an integral with variable bounds
#fundamental theorem of calculus #chain rule #limits
Riemann integral
L1
L1
Ex.395The largest rectangle in a semicircle
#optimisation #geometry #area
Optimisation and curve sketching
High school
High school
Ex.396The most economical can
#optimisation #cylinder #applications
Optimisation and curve sketching
High school
High school
Ex.397The ratio test
#ratio test #d'Alembert #factorials
Convergence tests
L1
L1
Ex.398Three ways to fail to be differentiable
#differentiability #corner #vertical tangent #oscillation
Differentiable functions
L1
L1
Ex.399Trigonometric integrals by linearisation
#trigonometric integrals #linearisation
Integration techniques
L1
L1
Ex.435Critical points of a polynomial
#critical points #polynomial #Hessian
Optimisation in several variables
L1
L1
Ex.436Summary of constrained optimization
#summary #constrained optimization
Optimisation in several variables
High school
High school
Ex.517Separable integrand on a square
#separability #product of integrals #square
Multiple integrals
High school
High school
Ex.518Summary of multiple integrals
#summary #multiple integrals
Multiple integrals
L1
L1
Ex.519Double integral over a triangular domain
#triangle #double integral #bounds
Multiple integrals
High school
High school
Ex.520Double integral under a parabola
#double integral #type-I domain #parabola
Multiple integrals
High school
High school
Ex.521A symmetric triangle integral
#symmetry #triangle #double integral
Multiple integrals
High school
High school
Ex.522Directional derivative
#directional derivative #gradient #direction
Partial derivatives
High school
High school
Ex.523Euler's theorem for homogeneous functions
#homogeneous functions #Euler #partial derivatives
Partial derivatives
L1
L1
Ex.524The gradient
#gradient #vector #direction
Partial derivatives
High school
High school
Ex.579The binomial expansion of √(1 + x)
#Taylor expansion #binomial series #square root
Taylor expansions
L1
L1
Ex.580The Taylor expansion of arctan by integrating
#Taylor expansion #integration of expansions #arctangent
Taylor expansions
L1
L1
Ex.591The divergence
#divergence #vector field #computation
Vector calculus
L1
L1
Ex.592Closed paths in conservative fields
#closed path #conservative #zero integral
Vector calculus
L1
L1
Ex.593Line integral of a scalar field
#line integral #scalar field #parametrization
Vector calculus
L1
L1
Ex.616A function with zero derivative is constant
#consequence #constant function #zero derivative
Mean Value Theorem
L1
L1
Ex.617Sign of the derivative and monotonicity
#monotonicity #derivative sign #consequence
Mean Value Theorem
L1
L1
Ex.639Cauchy's equation on the rationals
#Cauchy equation #additive function #rationals
Functional equations
M1
M1
Ex.662Equivalence between $\sum u_n$ and $\sum\ln(1+u_n)$
#equivalence #logarithm #ratio test #convergence
Convergence tests
L1
L1
Ex.698A recursive sequence converging to 2
#sequences #monotone convergence
Sequences
L1
L1
Ex.700Convergence of the series of 1/n²
#series #comparison
Convergence tests
L1
L1
Ex.702Divergence of the harmonic series
#series #divergence
Convergence tests
L1
L1
Ex.707Cosine to the power 1/x²
#exponential form #Taylor expansion #indeterminate forms
Limits
L2
L2
Ex.708Difference of two infinite reciprocals
#infinity minus infinity #Taylor expansion
Limits
L2
L2
Ex.709Epsilon-delta for x squared
#epsilon-delta #definition of limit #restricting delta
Limits
L1
L1
Ex.710The gap between two exponentials at infinity
#exponential #limits at infinity #change of variable
Limits
L1
L1
Ex.711The logarithm is slower than any power
#growth comparison #logarithm #inequalities
Limits
L1
L1
Ex.712(1 + sin x)^(1/x)
#exponential form #indeterminate forms #composition
Limits
L1
L1
Ex.713Polynomials versus the exponential
#growth comparison #exponential #squeeze theorem
Limits
L1
L1
Ex.714sin x − x cos x over x cubed
#Taylor expansion #little-o notation #cancellation
Limits
L2
L2
Ex.715sin(1/x) has no limit at 0
#sequences #non-existence of limits
Limits
L1
L1
Ex.716Tangent minus sine over x cubed
#trigonometric limits #factoring
Limits
L1
L1
Ex.717x ln x at zero, and x^x
#indeterminate forms #change of variable #logarithm
Limits
L1
L1
Ex.718x sin x has no limit at infinity
#sequences #non-existence of limits #infinite limits
Limits
L1
L1
Ex.719x times the Dirichlet function
#squeeze theorem #pathological functions #continuity
Limits
L1
L1
Ex.720A continuous function with rational values is constant
#intermediate value theorem #rational numbers #proof by contradiction
Intermediate Value Theorem
L1
L1
Ex.721Fixed point and a derivative bounded by one
#fixed points #mean value theorem #intermediate value theorem
Mean Value Theorem
L2
L2
Ex.722A recurrent sequence and its fixed point
#fixed point #contraction #mean value theorem
Sequences
L2
L2
Ex.723A reduction formula
#reduction formula #integration by parts #recurrence
Integration techniques
L1
L1
Ex.724Cancellation revealed by a Taylor expansion
#Taylor expansion #equivalents #cancellation
Convergence tests
L1
L1
Ex.725A telescoping sum that proves ζ(2) converges
#telescoping #comparison #zeta function
Convergence tests
L1
L1
Ex.726Absolute, conditional or no convergence
#absolute convergence #alternating series #p-series
Convergence tests
L1
L1
Ex.727Solving an integral equation
#integral equations #fundamental theorem #differential equation
Riemann integral
L1
L1
Ex.728Completing the square before integrating
#rational functions #completing the square #arctangent
Integration techniques
L1
L1
Ex.729An integral singular at both endpoints
#improper integrals #substitution #arcsine
Improper integrals
L1
L1
Ex.730Binet's formula for the Fibonacci numbers
#Fibonacci #golden ratio #linear recurrence #limits
Sequences
L1
L1
Ex.731Classifying a critical point when f″(0) = 0
#local extrema #Taylor expansion #second derivative test
Taylor expansions
L1
L1
Ex.732Comparing the harmonic sum with the logarithm
#integral comparison #harmonic series #monotonicity
Riemann integral
L1
L1
Ex.733Lagrange multipliers on a circle
#Lagrange #multipliers #equality constraint
Optimisation in several variables
L2
L2
Ex.734Optimisation under a linear constraint
#Lagrange #linear constraint #minimum
Optimisation in several variables
L2
L2
Ex.735Convergence to the golden ratio
#golden ratio #contraction #fixed point
Sequences
L1
L1
Ex.736cos x = 1/x has infinitely many solutions
#intermediate value theorem #infinitely many solutions
Intermediate Value Theorem
L1
L1
Ex.737Counting solutions depending on a parameter
#intermediate value theorem #monotonicity #parameters
Intermediate Value Theorem
L1
L1
Ex.738Integrating by parts twice and coming back
#integration by parts #exponential #trigonometric
Integration techniques
L1
L1
Ex.739Darboux sums of the Dirichlet function
#Riemann integrability #Darboux sums #density
Riemann integral
L1
L1
Ex.755Which is larger, e^π or π^e?
#optimisation #inequalities #logarithm
Optimisation and curve sketching
L1
L1
Ex.756tan x = x on each branch
#intermediate value theorem #tangent #monotonicity
Intermediate Value Theorem
L1
L1
Ex.757A linear first-order equation with polynomial right-hand side
#linear equation #homogeneous solution #particular solution
First-order equations
L1
L1
Ex.758Newton's law of cooling
#Newton cooling #separable equation #modelling
First-order equations
L1
L1
Ex.759Variation of constants
#variation of constants #linear equation #integration
First-order equations
L2
L2
Ex.760Functions whose derivative is proportional to themselves
#exponential function #differential equation #functional equations
Differentiable functions
L1
L1
Ex.761Heron's method for √2
#Newton's method #quadratic convergence #square roots
Sequences
L1
L1
Ex.762If f′ tends to ℓ, then f(x+1) − f(x) tends to ℓ
#mean value theorem #limits at infinity
Mean Value Theorem
L1
L1
Ex.763Integrating ln and arctan by parts
#integration by parts #logarithm #arctangent
Integration techniques
L1
L1
Ex.764L'Hôpital's rule, twice
#L'Hôpital's rule #indeterminate forms
Differentiable functions
L1
L1
Ex.765A diagonalizable system
#diagonalization #eigenvalues #eigenvectors
Linear systems
L2
L2
Ex.766A system with three distinct eigenvalues
#eigenvalues #eigenvectors #general solution
Linear systems
L2
L2
Ex.767Derivative of the matrix exponential
#matrix exponential #derivative #series
Linear systems
L3
L3
Ex.768Properties of the matrix exponential
#matrix exponential #commuting matrices #derivative
Linear systems
L3
L3
Ex.769Double integral in polar coordinates
#polar coordinates #double integral #Jacobian
Multiple integrals
L2
L2
Ex.770Triple integral over a simplex
#triple integral #iterated integration #simplex
Multiple integrals
L2
L2
Ex.771The chain rule for a composition
#chain rule #composition #directional derivative
Partial derivatives
L2
L2
Ex.772Power series of ln(1+x) and arctan x
#integration term by term #Leibniz series #ln 2
Power series
L2
L2
Ex.773Radius of convergence, four examples
#radius of convergence #ratio test #endpoints
Power series
L2
L2
Ex.774The sine of an alternating term
#alternating series #odd function #conditional convergence
Convergence tests
L1
L1
Ex.775Solving y″ = −y with a power series
#differential equations #power series #cosine #recurrence
Power series
L2
L2
Ex.776Summing power series in closed form
#power series #differentiation term by term #closed form
Power series
L2
L2
Ex.777Supremum, infimum, limsup and liminf
#supremum #limsup #liminf #bounded sequences
Sequences
L1
L1
Ex.778Approximating e with a guaranteed error
#Taylor-Lagrange #error bound #number e
Taylor expansions
L1
L1
Ex.779Alternating series and the error bound
#alternating series test #remainder #error estimate
Convergence tests
L1
L1
Ex.780The area of an ellipse
#trigonometric substitution #area #ellipse
Integration techniques
L1
L1
Ex.781The Cauchy–Schwarz inequality for integrals
#inequalities #Cauchy-Schwarz #quadratic polynomial
Riemann integral
L1
L1
Ex.782The closest point on a parabola
#optimisation #distance #parabola
Optimisation and curve sketching
L1
L1
Ex.783Differentiating under the integral sign
#parameter integrals #Gamma function #factorial
Improper integrals
M1
M1
Ex.784The hierarchy of growth
#growth comparison #ratio test #factorial
Sequences
L1
L1
Ex.785The integral test and Bertrand series
#integral test #Bertrand series #logarithm
Convergence tests
L1
L1
Ex.786Using the substitution x → a + b − x
#symmetry #substitution #trigonometric integrals
Integration techniques
L1
L1
Ex.787The limit comparison test
#limit comparison #equivalents #Taylor expansion
Convergence tests
L1
L1
Ex.788The limit of the derivative theorem
#mean value theorem #derivative at a point #extension
Mean Value Theorem
L2
L2
Ex.789The n-th derivative of 1/(x² − 1)
#higher derivatives #partial fractions
Differentiable functions
L1
L1
Ex.790The n-th derivative of x² e^x
#higher derivatives #Leibniz formula #induction
Differentiable functions
L1
L1
Ex.791The root test
#root test #Cauchy #exponential limits
Convergence tests
L1
L1
Ex.792Three cases of order-two recurrences
#linear recurrence #double root #complex roots
Sequences
L1
L1
Ex.793Two alternating series with logarithms
#alternating series test #monotonicity #conditional convergence
Convergence tests
L1
L1
Ex.794Line integral of a differential form
#line integral #differential form #parametrization
Vector calculus
L2
L2
Ex.795When L'Hôpital's rule gives no answer
#L'Hôpital's rule #counterexamples #squeeze theorem
Differentiable functions
L1
L1
Ex.796An equivalent of e^(sin x) − e^(tan x)
#equivalents #expansions #difference of functions
Taylor expansions
L2
L2
Ex.797A one-sided bound for √(1 + x) from the Lagrange remainder
#Taylor-Lagrange #inequality #remainder sign
Taylor expansions
L1
L1
Ex.798Computing ∫₀¹ sin(x²) dx with a series
#integration of expansions #numerical approximation #alternating series
Taylor expansions
L2
L2
Ex.840A fast series for ln 2
#logarithm #series #numerical approximation #speed of convergence
Taylor expansions
L2
L2
Ex.857Boundary classification
#boundary #classification #circle
Optimisation in several variables
L1
L1
Ex.858Distance from point to plane
#Lagrange #distance #plane
Optimisation in several variables
L1
L1
Ex.859Two critical points and their nature
#critical points #saddle #maximum
Optimisation in several variables
L1
L1
Ex.860Lagrange multipliers on a circle
#Lagrange #circle #extrema
Optimisation in several variables
L1
L1
Ex.861Optimization under a linear constraint
#Lagrange #linear constraint #minimum
Optimisation in several variables
L1
L1
Ex.862Maximizing a product
#Lagrange #product #constraint
Optimisation in several variables
L1
L1
Ex.863A unique critical point that is a saddle
#saddle #unique critical #Hessian
Optimisation in several variables
L1
L1
Ex.864A discontinuous function with bounded values
#discontinuity #bounded #counterexample
Continuity
L1
L1
Ex.865Discontinuity via two paths
#discontinuity #paths #limits
Continuity
L1
L1
Ex.866Norm and distance are continuous
#norm #distance #continuity
Continuity
L1
L1
Ex.867Continuity of a piecewise function
#piecewise #continuity #xy
Continuity
L1
L1
Ex.868Continuity via polar coordinates
#polar #continuity #bound
Continuity
L1
L1
Ex.869Continuous extension
#continuous extension #removable singularity
Continuity
High school
High school
Ex.870cosh x ≤ e^(x²/2)
#series comparison #hyperbolic functions #inequalities
Taylor expansions
L2
L2
Ex.956Finite difference formulas for f′
#finite differences #numerical differentiation #error estimate
Taylor expansions
L3
L3
Ex.985Derivatives of high order at 0 without differentiating
#Taylor expansion #uniqueness #high order derivatives
Taylor expansions
L2
L2
Ex.1047Approximating x by a combination of sines
#undetermined coefficients #Taylor expansion #Richardson extrapolation
Taylor expansions
L2
L2
Ex.1071Boundary effects
#boundary #integration bounds #evaluation
Multiple integrals
L1
L1
Ex.1072Cartesian vs polar computation
#polar vs cartesian #comparison #disk
Multiple integrals
L1
L1
Ex.1073Double integral over an annulus
#polar coordinates #annulus #angular integral
Multiple integrals
L1
L1
Ex.1074Double integral over a disk tangent at origin
#polar coordinates #circle offset #integral
Multiple integrals
L1
L1
Ex.1075Double integral over a disk in polar coordinates
#polar coordinates #disk #Jacobian
Multiple integrals
L1
L1
Ex.1076Polar integral over a half-disk
#polar #half-disk #integral
Multiple integrals
L1
L1
Ex.1077Mass of a cube with variable density
#triple integral #density #cube
Multiple integrals
L1
L1
Ex.1078Triple integral over a simplex
#triple integral #simplex #iterated
Multiple integrals
L1
L1
Ex.1079General chain rule
#chain rule #composition #multivariable
Partial derivatives
L1
L1
Ex.1080Clairaut's theorem
#Clairaut #Schwarz #mixed partials
Partial derivatives
L1
L1
Ex.1081Saddle point
#saddle point #Hessian #classification
Optimisation in several variables
L1
L1
Ex.1082The differential
#differential #linearization #approximation
Partial derivatives
L1
L1
Ex.1083Second-order partial derivatives
#higher order #Schwarz #second derivatives
Partial derivatives
L1
L1
Ex.1084The Laplacian
#Laplacian #harmonic functions #second derivatives
Partial derivatives
L1
L1
Ex.1085Finding critical points
#critical points #stationary #gradient
Partial derivatives
L1
L1
Ex.1086Schwarz's theorem and a counterexample
#Schwarz theorem #mixed partials #symmetry
Partial derivatives
L1
L1
Ex.1087Tangent plane to a graph
#tangent plane #differentiability #graph
Partial derivatives
L1
L1
Ex.1175Convergence of three series via expansions
#series #equivalents #comparison
Taylor expansions
L2
L2
Ex.1181Expanding x/(eˣ − 1)
#Taylor expansion #quotient #Bernoulli numbers #undetermined coefficients
Taylor expansions
L2
L2
Ex.1182Taylor series of the solution of y′ = x + y²
#differential equation #Taylor expansion #successive derivatives
Taylor expansions
L2
L2
Ex.1183The symmetric second difference
#Taylor-Young #second derivative #counterexample
Taylor expansions
L2
L2
Ex.1198The curl
#curl #vector field #rotation
Vector calculus
L1
L1
Ex.1201Flux through a flat surface
#flux #plane surface #dot product
Vector calculus
L1
L1
Ex.1202An application of Green's theorem
#Green #application #area
Vector calculus
L1
L1
Ex.1203Area using Green's theorem
#area #Green #closed curve
Vector calculus
L1
L1
Ex.1204Green with symmetry
#Green #symmetry #shortcut
Vector calculus
L1
L1
Ex.1210Conservative vector field
#conservative #gradient #path independence
Vector calculus
L1
L1
Ex.1211Fundamental theorem for line integrals
#fundamental theorem #line integral #potential
Vector calculus
L1
L1
Ex.1212Line integral along a circle
#line integral #circle #parametrization
Vector calculus
L1
L1
Ex.1213Line integral on a piecewise path
#piecewise path #line integral #work
Vector calculus
L1
L1
Ex.1214Work done by a force field
#work #line integral #vector field
Vector calculus
L1
L1
Ex.1215Testing whether a field is conservative
#conservative #curl #test
Vector calculus
L1
L1
Ex.1258f(x + f(y)) = f(x) + y on the rationals
#injectivity #involution #Cauchy equation on Q
Functional equations
Olympiad
Olympiad
Ex.1268An arctangent inequality
#mean value theorem #inequality #arctan
Mean Value Theorem
L1
L1
Ex.1269An exponential inequality
#mean value theorem #inequality #exponential
Mean Value Theorem
L1
L1
Ex.1270A logarithm inequality via the MVT
#mean value theorem #inequality #logarithm
Mean Value Theorem
L1
L1
Ex.1271Bounding $f(b)-f(a)$
#MVT #bound #interval
Mean Value Theorem
L1
L1
Ex.1272Small-increment estimate
#MVT #small increments #estimate
Mean Value Theorem
L1
L1
Ex.1273Bounded derivative implies Lipschitz
#Lipschitz #bounded derivative #MVT
Mean Value Theorem
L1
L1
Ex.1274Rolle and the sign of $f'$
#Rolle #sign change #derivative
Mean Value Theorem
L1
L1
Ex.1275Rolle on a polynomial with three roots
#Rolle #multiple roots #derivative
Mean Value Theorem
L1
L1
Ex.1276Uniqueness of a root
#Rolle #uniqueness #contrapositive
Mean Value Theorem
L1
L1
Ex.1314f(x)f(y) − f(xy) = x + y
#functional equation #substitution #polynomial identity
Functional equations
M1
M1
Ex.1393Three logarithmic series
#logarithmic series #telescoping #conditional convergence
Find the sum
L1
L1
Ex.1394Six numeric series: convergence study
#ratio test #root test #comparison #absolute convergence
Convergence tests
L1
L1
Ex.1395Three convergent series with explicit sums
#geometric series #telescoping #arctangent identity #d'Alembert
Find the sum
L1
L1
Ex.1396Three series with computed sums
#geometric series #telescoping #arctan identity
Find the sum
L1
L1
Ex.1444Continuity of a piecewise function with an essential singularity
#piecewise #continuity #exponential singularity
Continuity
L1
L1
Ex.1464How fast does (1 + x)^(1/x) reach e?
#number e #Taylor expansion #order of convergence
Limits
L2
L2
Ex.1465Where power means lead
#power mean #geometric mean #logarithm #expansions
Limits
L2
L2
Ex.1466Reciprocal squares of x and tangent
#Taylor expansion #infinity minus infinity #tangent
Limits
L2
L2
Ex.1467The Dirichlet function has no limit anywhere
#sequences #density #non-existence of limits #pathological functions
Limits
L1
L1
Ex.1468A recurrence with a double fixed point
#fixed points #monotone sequences #parameter dependence
Sequences
L1
L1
Ex.1469A functional equation solved with derivatives
#functional equations #differentiation #polynomials
Differentiable functions
Olympiad
Olympiad
Ex.1470The series of 1/(ln n)^(ln n)
#comparison test #exponential form #logarithm
Convergence tests
L1
L1
Ex.1471Adjacent sequences and the irrationality of e
#adjacent sequences #number e #irrationality
Sequences
L1
L1
Ex.1472Bounds for n! by comparing with an integral
#factorial #integral comparison #Stirling
Riemann integral
L1
L1
Ex.1473Cauchy's condensation test
#condensation test #p-series #monotone terms
Convergence tests
M1
M1
Ex.1474A Cauchy product that diverges
#Cauchy product #conditional convergence #Mertens theorem
Convergence tests
L1
L1
Ex.1475Cauchy-Schwarz inequality via optimisation
#Cauchy-Schwarz #optimisation #proof
Optimisation in several variables
Olympiad
Olympiad
Ex.1476Constrained extrema and a compactness argument
#Lagrange #compactness #global maximum
Optimisation in several variables
L3
L3
Ex.1477Maximizing a product under a linear constraint
#Lagrange #product #linear constraint
Optimisation in several variables
L3
L3
Ex.1478Least squares as an optimisation problem
#least squares #optimisation #regression
Optimisation in several variables
L3
L3
Ex.1479Continuous injective functions are monotone
#intermediate value theorem #injectivity #monotonicity
Intermediate Value Theorem
L2
L2
Ex.1480A convex function and its asymptote
#convexity #slopes #asymptotes
Differentiable functions
L1
L1
Ex.1481Convexity forces the derivative to be continuous
#convexity #Darboux theorem #monotone functions #continuity
Differentiable functions
L1
L1
Ex.1489An integrating factor
#integrating factor #linear equation #exact equation
First-order equations
L2
L2
Ex.1490The generating function of Fibonacci numbers
#generating function #Fibonacci #partial fractions #radius of convergence
Power series
L2
L2
Ex.1491How many solutions does ln x = ax have?
#intermediate value theorem #parameters #logarithm
Intermediate Value Theorem
L1
L1
Ex.1492Integrability with one wild point
#Riemann integrability #Darboux criterion #oscillation
Riemann integral
L1
L1
Ex.1493Limits of integrals depending on n
#limits of integrals #squeeze theorem #integration by parts
Riemann integral
L1
L1
Ex.1494Complex eigenvalues
#complex eigenvalues #real solutions #coefficients ind�termin�s
Linear systems
L3
L3
Ex.1495Matrix exponential with a nilpotent part
#matrix exponential #nilpotent #Jordan form
Linear systems
L3
L3
Ex.1496A linear system with a second member
#second member #variation of constants #matrix exponential
Linear systems
L3
L3
Ex.1497Affine change of variables
#change of variables #Jacobian #parallelogram
Multiple integrals
L2
L2
Ex.1498Is a function $C^1$ on $\mathbb{R}^2$?
#continuity #partial derivatives #class C1
Partial derivatives
L2
L2
Ex.1499A series of functions
#series of functions #normal convergence #pointwise convergence
Power series
L2
L2
Ex.1500Resonance with a double root
#resonance #double root #particular solution #identification
Linear equations of order two
L2
L2
Ex.1501A second-order equation with a polynomial right-hand side
#particular solution #identification #polynomial right-hand side
Linear equations of order two
L2
L2
Ex.1502The sequence sin(n) does not converge
#divergence #trigonometry #subsequences #contradiction
Sequences
L1
L1
Ex.1503Speed of convergence
#speed of convergence #fixed point #mean value theorem
Sequences
L3
L3
Ex.1504The alternating harmonic sums converge
#adjacent subsequences #alternating series #ln 2
Sequences
L2
L2
Ex.1505The arithmetic–geometric mean
#adjacent sequences #AM-GM #means
Sequences
L1
L1
Ex.1506The generating function of central binomial coefficients
#binomial series #generalised binomial coefficients #radius of convergence
Power series
L2
L2
Ex.1507The Cauchy criterion
#Cauchy sequences #completeness #series
Sequences
L1
L1
Ex.1508A ratio test that fails, and how to conclude
#ratio test #central binomial coefficients #induction #Stirling
Convergence tests
L1
L1
Ex.1509The Cesàro lemma
#Cesaro mean #epsilon-N #averages
Sequences
L1
L1
Ex.1510The Euler-Mascheroni constant as a series
#Euler-Mascheroni constant #harmonic numbers #telescoping #logarithm
Find the sum
L1
L1
Ex.1511e^x e^y = e^(x+y) from power series
#Cauchy product #exponential series #binomial theorem
Power series
L2
L2
Ex.1512The horizontal chord of length one half
#intermediate value theorem #chords #clever auxiliary function
Intermediate Value Theorem
L2
L2
Ex.1513The longest ladder around a corner
#optimisation #trigonometry #geometry #applications
Optimisation and curve sketching
L2
L2
Ex.1514From the ratio to the n-th root
#Cesaro lemma #n-th root #ratio #Stirling
Sequences
L2
L2
Ex.1515The ratio test with a parameter
#ratio test #Stirling formula #critical case
Convergence tests
L1
L1
Ex.1516Monotonicity of (1 + 1/n)^n
#number e #AM-GM inequality #monotone convergence
Sequences
L1
L1
Ex.1517The tangent half-angle substitution
#Weierstrass substitution #trigonometric integrals #rational functions
Integration techniques
L1
L1
Ex.1518Divergence theorem (Ostrogradsky)
#divergence theorem #flux #surface
Vector calculus
L3
L3
Ex.1519Green-Riemann formula
#Green-Riemann #line integral #double integral
Vector calculus
L3
L3
Ex.1520Stokes' theorem
#Stokes #curl #surface integral
Vector calculus
L3
L3
Ex.1521Wallis integrals
#Wallis integrals #recurrence #asymptotics #pi
Integration techniques
L1
L1
Ex.1522Differentiable but not continuously differentiable
#differentiability #continuity of the derivative #counterexamples
Differentiable functions
L1
L1
Ex.1523An expansion of order 2 without a second derivative
#Taylor-Young converse #counterexample #differentiability
Taylor expansions
L3
L3
Ex.1524A rational approximation of cosine
#Padé approximant #undetermined coefficients #approximation
Taylor expansions
L2
L2
Ex.1525Expanding the inverse of x + x³
#Taylor expansion #inverse function #fixed point iteration
Taylor expansions
L2
L2
Ex.1557Bounding f′ by f and f″ (Landau's inequality)
#Taylor-Lagrange #inequalities #optimisation of a parameter
Taylor expansions
M1
M1
Ex.1566Central binomial coefficients from (1 − 4x)^(−1/2)
#binomial series #generalised binomial coefficients #central binomial
Taylor expansions
L2
L2
Ex.1591Cauchy–Schwarz via optimization
#Cauchy-Schwarz #optimization #inequality
Optimisation in several variables
L1
L1
Ex.1592Optimization under inequality constraints
#inequality constraint #KKT #boundary
Optimisation in several variables
L1
L1
Ex.1593Classifying a 3D critical point
#Hessian #3D #saddle
Optimisation in several variables
M1
M1
Ex.1594Three-variable saddle point
#three variables #saddle #Hessian
Optimisation in several variables
L1
L1
Ex.1595Shortest distance from a point to a line
#Lagrange #distance #line
Optimisation in several variables
L1
L1
Ex.1596Distance from origin to an ellipsoid
#Lagrange #ellipsoid #distance
Optimisation in several variables
L1
L1
Ex.1597Lagrange with exponential objective
#Lagrange #exponential #constrained maximum
Optimisation in several variables
L1
L1
Ex.1598Global extrema on a compact set
#compactness #global extrema #boundary
Optimisation in several variables
L1
L1
Ex.1599Maximizing a volume with fixed perimeter
#Lagrange #geometric optimization #volume
Optimisation in several variables
L1
L1
Ex.1600Constrained quadratic form
#quadratic form #Rayleigh quotient #eigenvalues
Optimisation in several variables
M1
M1
Ex.1601Image of a compact set under a continuous function
#compactness #continuous image #extrema
Continuity
L1
L1
Ex.1602A function that is not continuous despite radial limits
#continuity #radial limit #counterexample
Continuity
L1
L1
Ex.1657e is irrational
#Taylor-Lagrange #irrationality #exponential
Taylor expansions
L2
L2
Ex.1783Centre of gravity of a planar region
#centroid #parabola #uniform density
Multiple integrals
L1
L1
Ex.1784Linear change of variables
#change of variables #Jacobian #triangle
Multiple integrals
L1
L1
Ex.1785Change of variables to simplify the integrand
#change of variables #Jacobian #triangular
Multiple integrals
L1
L1
Ex.17863D change of variables
#change of variables #3D #Jacobian
Multiple integrals
L1
L1
Ex.1787Volume using cylindrical coordinates
#cylindrical coordinates #volume #cone
Multiple integrals
L1
L1
Ex.1788The Fubini trick
#Fubini #double integral #iterated
Multiple integrals
L1
L1
Ex.1789Polar region between two circles
#polar #circle #region between circles
Multiple integrals
L1
L1
Ex.1790Spherical coordinates with a radial integrand
#spherical coordinates #radial #integration
Multiple integrals
L1
L1
Ex.1791Volume of a sphere by triple integral
#spherical coordinates #volume #sphere
Multiple integrals
L1
L1
Ex.1792Triple integral over a cube
#cube #triple integral #iterated
Multiple integrals
L1
L1
Ex.1793Triple integral over a simplex with cube
#simplex #triple integral #iterated
Multiple integrals
L1
L1
Ex.1794Triple integral with sine
#triple integral #sine #simplex
Multiple integrals
L1
L1
Ex.1795Differentiability of a piecewise function
#differentiability #continuity #piecewise
Partial derivatives
L1
L1
Ex.1796The Hessian matrix
#Hessian #second derivatives #quadratic form
Optimisation in several variables
L1
L1
Ex.1797Implicit differentiation
#implicit differentiation #Implicit Function Theorem
Partial derivatives
L1
L1
Ex.1798Inverse function theorem (illustration)
#inverse function theorem #Jacobian #local invertibility
Partial derivatives
M1
M1
Ex.1799Jacobian of a composition
#Jacobian #composition #chain rule
Partial derivatives
L1
L1
Ex.1800Level curves and gradients
#level curves #gradient #orthogonality
Partial derivatives
L1
L1
Ex.1801Mean value inequality
#mean value #inequality #gradient
Partial derivatives
M1
M1
Ex.1802Taylor expansion of order 2
#Taylor expansion #multivariable #approximation
Partial derivatives
L1
L1
Ex.1883Asymptotic expansion of (1 + 1/n)ⁿ
#asymptotic expansion #sequences #composition of expansions
Taylor expansions
L2
L2
Ex.1903Gauss divergence theorem for a cylinder
#divergence #cylinder #flux
Vector calculus
M1
M1
Ex.1904The divergence theorem
#divergence theorem #flux #surface
Vector calculus
L1
L1
Ex.1905Flux through a cube
#divergence #cube #flux
Vector calculus
L1
L1
Ex.1908Divergence theorem with variable divergence
#divergence #sphere #variable density
Vector calculus
M1
M1
Ex.1910Recovering a field from its curl (2D)
#Green #curl #reconstruction
Vector calculus
M1
M1
Ex.1911Green's reciprocity
#Green #reciprocity #symmetry
Vector calculus
M1
M1
Ex.1912Green's theorem on an annulus
#Green #annulus #inner boundary
Vector calculus
L1
L1
Ex.1913Green's theorem
#Green theorem #curl #region
Vector calculus
L1
L1
Ex.1914Green's theorem with a general field
#Green #general field #region
Vector calculus
L1
L1
Ex.1917Area of a region using line integral
#line integral #area #polar
Vector calculus
L1
L1
Ex.1918Conservative fields in 3D
#conservative #3D #potential
Vector calculus
M1
M1
Ex.1919Finding a potential function
#potential #conservative #integration
Vector calculus
L1
L1
Ex.1925Stokes with a hemisphere
#Stokes #hemisphere #closed curve
Vector calculus
L1
L1
Ex.1926Stokes' theorem
#Stokes theorem #curl #surface
Vector calculus
L1
L1
Ex.1943An inequality as a functional equation (IMO 1977)
#IMO #natural numbers #induction #monotonicity
Functional equations
Olympiad
Olympiad
Ex.1953Cauchy's extended mean value theorem
#Cauchy #extended MVT #ratio
Mean Value Theorem
L1
L1
Ex.1954Darboux's theorem on derivatives
#Darboux #intermediate value #derivatives
Mean Value Theorem
M1
M1
Ex.1955Second-order Taylor formula
#Taylor #MVT #second order
Mean Value Theorem
L1
L1
Ex.1956Iterated Rolle
#Rolle #higher derivatives #iterated
Mean Value Theorem
L1
L1
Ex.1979Additive functions with f(x²) = f(x)²
#additive function #monotonicity #Cauchy equation
Functional equations
Olympiad
Olympiad
Ex.1980Solving f(f(n)) = n + 2 on the positive integers
#functional equation #integers #parity #recursion
Functional equations
Olympiad
Olympiad
Ex.1981f(f(x) + y) = 2x + f(f(y) − x)
#surjectivity #substitution #linear functions
Functional equations
Olympiad
Olympiad
Ex.1982IMO 1968: f(x + a) = ½ + √(f(x) − f(x)²)
#IMO 1968 #periodicity #squaring #completing squares
Functional equations
Olympiad
Olympiad
Ex.1983IMO 2010: f(⌊x⌋y) = f(x)⌊f(y)⌋
#IMO 2010 #floor function #functional equation #case analysis
Functional equations
Olympiad
Olympiad
Ex.2034Four series involving logarithms
#root test #comparison #equivalent #logarithm
Convergence tests
L1
L1
Ex.2067Tangent of sine minus sine of tangent
#Taylor expansion #composition #high order #challenge
Limits
L2
L2
Ex.2068An absolute value integral with a parameter
#absolute value #parameters #trigonometry
Riemann integral
Olympiad
Olympiad
Ex.2069The root of x^n + x = 1
#implicit sequences #monotonicity #intermediate value theorem
Sequences
L1
L1
Ex.2070Rayleigh quotient and eigenvalues
#Rayleigh quotient #eigenvalues #optimisation on the sphere
Optimisation in several variables
Olympiad
Olympiad
Ex.2071The AM-GM inequality via optimisation
#inequality #AM-GM #optimisation #proof
Optimisation in several variables
Olympiad
Olympiad
Ex.2072Convergence of the series of sin(n)/n
#Abel summation #Dirichlet test #trigonometric series
Convergence tests
M1
M1
Ex.2073Interlacing an arithmetic and a geometric progression
#arithmetic progression #geometric progression #inequalities #construction
Sequences
Olympiad
Olympiad
Ex.2074Rearranging a conditionally convergent series
#conditional convergence #rearrangement #Riemann theorem #ln 2
Find the sum
L1
L1
Ex.2075Legendre polynomials have all their roots in (-1, 1)
#Rolle's theorem #polynomials #induction #Legendre
Mean Value Theorem
L2
L2
Ex.2076Superposition of right-hand sides
#superposition #double root #polynomial right-hand side
Linear equations of order two
L2
L2
Ex.2077The integral of sin x / x
#improper integrals #conditional convergence #integration by parts
Improper integrals
M1
M1
Ex.2078An integral that equals its own mirror image
#symmetry #substitution #logarithm #tangent
Integration techniques
Olympiad
Olympiad
Ex.2079The logistic sequence decays like 1/n
#Cesaro lemma #asymptotic equivalent #recurrence
Sequences
L2
L2
Ex.2080The popcorn function
#continuity #rationals #density #pathological functions
Continuous and discontinuous functions
L2
L2
Ex.2081Reciprocals of integers with no digit 9
#comparison test #counting #harmonic series #Kempner
Convergence tests
L1
L1
Ex.2082A series with an equivalent that converges but does not
#equivalents #alternating series #counterexample
Convergence tests
M1
M1
Ex.2083A smooth function that is not its Taylor series
#flat function #smooth #analytic #counterexample
Taylor expansions
M1
M1
Ex.2115Two equality constraints
#Lagrange #multiple constraints #KKT
Optimisation in several variables
M1
M1
Ex.2183Gaussian integral over $\mathbb{R}^2$
#Gaussian #polar #improper integral
Multiple integrals
M1
M1
Ex.2184Jacobian of a nonlinear transformation
#Jacobian #nonlinear #change of variables
Multiple integrals
M1
M1
Ex.2185Harmonic conjugate
#harmonic conjugate #Cauchy-Riemann #complex
Partial derivatives
M1
M1
Ex.2186Partial derivatives exist but function is not differentiable
#differentiability #counterexample #partial derivatives
Partial derivatives
L1
L1
Ex.2223Stokes on a cone
#Stokes #cone #surface
Vector calculus
M1
M1
Ex.2224Stokes on a torus (statement)
#Stokes #torus #closed surface
Vector calculus
M1
M1
Ex.2238A strictly increasing f with f(f(n)) = 3n
#functional equation #monotone #integers #ternary structure
Functional equations
Olympiad
Olympiad