Maths WorkshopMaths problems with full solutions

Optimisation and curve sketching

38 problems · Extrema and the study of functions.

Ex.15A fence along a river
#optimisation #area #quadratic function
Optimisation and curve sketching
High school
High school
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.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.755Which is larger, e^π or π^e?
#optimisation #inequalities #logarithm
Optimisation and curve sketching
L1
L1
Ex.782The closest point on a parabola
#optimisation #distance #parabola
Optimisation and curve sketching
L1
L1
Ex.1513The longest ladder around a corner
#optimisation #trigonometry #geometry #applications
Optimisation and curve sketching
L2
L2