Limits

problem
Favourite

Exponential Trend

Age

16 to 18

Challenge level

1 out of 3

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.
problem
Favourite

Spokes

Age

16 to 18

Challenge level

3 out of 3

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
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Favourite

There's a Limit

Age

14 to 18

Challenge level

1 out of 3

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?
problem
Favourite

Witch of Agnesi

Age

16 to 18

Challenge level

1 out of 3

Sketch the members of the family of graphs given by $y = a^3/(x^2+a^2)$ for $a=1, 2$ and $3$.
problem
Favourite

Converging Product

Age

16 to 18

Challenge level

2 out of 3

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?
problem
Favourite

Rain or Shine

Age

16 to 18

Challenge level

2 out of 3

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.
problem
Favourite

Discrete Trends

Age

16 to 18

Challenge level

2 out of 3

Find the maximum value of n to the power 1/n and prove that it is a maximum.
problem

Reciprocal Triangles

Age

16 to 18

Challenge level

2 out of 3

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.
problem

Lower Bound

Age

14 to 16

Challenge level

2 out of 3

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
problem

Squareflake

Age

16 to 18

Challenge level

2 out of 3

A finite area inside and infinite skin! You can paint the interior of this fractal with a small tin of paint but you could never get enough paint to paint the edge.