Episode 20: Calculus and fluid dynamics

This week the topic was calculus and differentiation. We talked to Florencia Tettamanti, who’s a mathematician working on fluid dynamics. We talked about how Flo uses calculus to study the motion of fluids like air and water, and what it’s like to be a research mathematician.

Interesting links:

• Basic differentiation, at s-cool
• Differential equations, at the University of Surrey website
• Fluid dynamics on Wikipedia
• NSF videos on Fluid Mechanics - YouTube playlist

Puzzle: If your function is given by y = x² - 6x + 13, what is the minimum value of y, and for which value of x does the function give this value?

Solution:

If you plot the points x=1, x=2, x=3 and x=4 you can clearly see the curve of this graph and that it seems to have a maximum at x=3, for which the value of y is 4. To see what the graph looks like, you can input the equation into Wolfram Alpha.

Another way to see this is to rearrange the equation: x²-6x+13 = (x-3)²+4, and by examining this equation we can see that this is just an x graph, shifted across by 3 and up by 4, so its turning point and hence the minimum will be at x=3 and y=4.

If you know how to use calculus, you can find the turning point more easily - if you differentiate x²-6x+13 you get 2x - 6, which will equal zero when x=3, and putting this value back into the original equation gives y=4.