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Broad Topics >

Calculus > Limits

##### Age 16 to 18

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

##### Age 16 to 18

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

##### Age 16 to 18

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Each week a company produces X units and sells p per cent of its
stock. How should the company plan its warehouse space?

##### Age 16 to 18 Challenge Level:

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.

##### Age 14 to 16 Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Find the equation from which to calculate the resistance of an
infinite network of resistances.

##### Age 16 to 18 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

##### Age 16 to 18 Challenge Level:

Find a connection between the shape of a special ellipse and an
infinite string of nested square roots.

##### Age 16 to 18 Challenge Level:

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.

##### Age 16 to 18 Challenge Level:

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

##### Age 14 to 18 Challenge Level:

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?

##### Age 16 to 18 Challenge Level:

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.

##### Age 14 to 16 Challenge Level:

Keep constructing triangles in the incircle of the previous triangle. What happens?

##### Age 16 to 18 Challenge Level:

Find the link between a sequence of continued fractions and the
ratio of succesive Fibonacci numbers.

##### Age 16 to 18 Challenge Level:

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)... ?

##### Age 16 to 18 Challenge Level:

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