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Broad Topics > Pre-Calculus and Calculus > Limits

Fractional Calculus II

Stage: 5

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.

Fractional Calculus I

Stage: 5

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.

Fractional Calculus III

Stage: 5

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

There's a Limit

Stage: 4 and 5 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?

Reciprocal Triangles

Stage: 5 Challenge Level:

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

Discrete Trends

Stage: 5 Challenge Level:

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

Squaring the Circle and Circling the Square

Stage: 4 Challenge Level:

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

Golden Eggs

Stage: 5 Challenge Level:

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

Witch of Agnesi

Stage: 5 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.

Squareflake

Stage: 5 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.

Production Equation

Stage: 5 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?

Triangle Incircle Iteration

Stage: 4 Challenge Level:

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .

Exponential Trend

Stage: 5 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.

Converging Product

Stage: 5 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)... ?

Over the Pole

Stage: 5 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.

Golden Fractions

Stage: 5 Challenge Level:

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

Resistance

Stage: 5 Challenge Level:

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

Spokes

Stage: 5 Challenge Level:

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

Rain or Shine

Stage: 5 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.