Maths WorkshopMaths problems with full solutions

Differentiation

118 problems · Derivatives, variations, convexity and inequalities from derivatives.

Ex.8Polynomial bounds for sine and cosine
#inequalities #monotonicity
Differentiable functions
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.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.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.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.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.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.109Distance to a line, without a formula
#optimisation #distance #line #quadratic function
Optimisation and curve sketching
High school
High school
Ex.116Least squares for a line through the origin
#least squares #regression #statistics #quadratic
Optimisation and curve sketching
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.203Derivative of a quotient
#derivative #quotient rule
Differentiable functions
High school
High school
Ex.211Rolle's theorem on a quartic
#Rolle's theorem #polynomials
Mean Value Theorem
L1
L1
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.343The logarithm lies below its tangent
#inequalities #derivatives
Differentiation
L1
L1
Ex.349Sine is 1-Lipschitz
#mean value theorem #inequalities
Mean Value Theorem
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.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.387Tangent lines through an external point
#tangent line #parabola #geometry
Differentiable functions
High school
High school
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.398Three ways to fail to be differentiable
#differentiability #corner #vertical tangent #oscillation
Differentiable functions
L1
L1
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.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.721Fixed point and a derivative bounded by one
#fixed points #mean value theorem #intermediate value theorem
Mean Value Theorem
L2
L2
Ex.731Classifying a critical point when f″(0) = 0
#local extrema #Taylor expansion #second derivative test
Taylor expansions
L1
L1
Ex.755Which is larger, e^π or π^e?
#optimisation #inequalities #logarithm
Optimisation and curve sketching
L1
L1
Ex.760Functions whose derivative is proportional to themselves
#exponential function #differential equation #functional equations
Differentiable functions
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.764L'Hôpital's rule, twice
#L'Hôpital's rule #indeterminate forms
Differentiable functions
L1
L1
Ex.778Approximating e with a guaranteed error
#Taylor-Lagrange #error bound #number e
Taylor expansions
L1
L1
Ex.782The closest point on a parabola
#optimisation #distance #parabola
Optimisation and curve sketching
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.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.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.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.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.1469A functional equation solved with derivatives
#functional equations #differentiation #polynomials
Differentiable functions
Olympiad
Olympiad
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.1513The longest ladder around a corner
#optimisation #trigonometry #geometry #applications
Optimisation and curve sketching
L2
L2
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.1657e is irrational
#Taylor-Lagrange #irrationality #exponential
Taylor expansions
L2
L2
Ex.1883Asymptotic expansion of (1 + 1/n)ⁿ
#asymptotic expansion #sequences #composition of expansions
Taylor expansions
L2
L2
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.2075Legendre polynomials have all their roots in (-1, 1)
#Rolle's theorem #polynomials #induction #Legendre
Mean Value Theorem
L2
L2
Ex.2083A smooth function that is not its Taylor series
#flat function #smooth #analytic #counterexample
Taylor expansions
M1
M1