Generally Geometric

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Exponential Trend

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Slide

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

Bend

Age 16 to 18 Challenge Level:

This solution comes from Andrei Lazanu ,Tudor Vianu National College, Bucharest, Romania.

I calculate the length of the stick in terms of $a$, $b$ and $\theta$. From the figure I observe that the length of the stick could be seen as the sum of two hypotenuses of two right-angled triangles. Its length is: $$l(\theta)= {a\over \sin \theta} + {b\over \cos \theta}.$$ Now, I have to calculate the minimum of this expression, in order to make the stick pass through the corner. For this, I calculate the derivative of $l(\theta)$ and equate it to $0$. I must say from the beginning that derivatives are not so familiar to me. For a minimum value of the length: $${\rm{d}l\over \rm{d}\theta} = {-a\cos \theta \over \sin^2 \theta} + {b\sin \theta \over \cos^2 \theta}=0.$$ So for a minimum value $a\cos^3 \theta = b\sin^3 \theta$ and $$\tan \theta = \left({a\over b}\right)^{1/3}.$$ Now, I have to calculate $\sin\theta$ and $\cos\theta$ as functions of $\tan\theta$. I know that: $$\cos x = {1\over \sqrt{1+\tan^2x}}$$ and$$\sin x = {\tan x \over \sqrt{1+\tan^2x}}$$ In the case of the problem, I have: $${1\over \cos \theta }= \sqrt{1+\left({a\over b}\right)^{2/3}}$$ and $${1\over \sin \theta } = \left({b\over a}\right)^{1/3}\sqrt{1+\left({a\over b}\right)^{2/3}}$$ So the minimum length is \eqalign{ {a\over \sin \theta} + {b\over \cos \theta} &=\left(a^{2/3}b^{1/3} + b\right)\sqrt{{a^{2/3}+b^{2/3}\over b^{2/3}}} \cr &= \left(a^{2/3} + b^{2/3}\right)^{3/2}}. The result is symmetric in $a$ and $b$.

If $a=65 \text{ cm}$ and $b=75 \text{ cm}$ then $65^{2/3}+ 75^{2/3}=16.16623563 + 17.78446652=33.95070215$ and $33.951^{3/2}=197.8213407$ so an object of about $197 \text{ cm}$ could be manoeuvred around the bend but it is not possible to manoeuvre a $200 \text {cm}$ object around this bend.