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Generally Geometric

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

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Bend

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

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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.

Ramping it Up

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

 

Stepped function

At a time T, a car mounts a curb of height $H$, and its front suspension undergoes a sudden change. At a time 2T, the car drops down the other side of the curb to $\frac{1}{4}H$. It is suggested that the graph of this process is as given as a function $f(t)$ above.

Try to work out the graph of $f $ '$(t)$ against $t$. What parts make sense and what parts don't? Could you alter the graph of $f(t)$ slightly to make the derivative function make more sense?

Next imagine that a car drives at a steady $20$ km h$^{-1}$ over a concrete block of height $10$cm and rectangular cross section of width $50$ cm. Make a mathematical model of the height of the front suspension over time. Be clear as to your assumptions and make a clear note of any possible refinements to your model.

What does your model say about the derivative of the height over time?

What is the maximum G-force the suspension of a car is likely to be subjected to in normal use?

On a physical note, think about the implications that this sort of modelling process might have on the design of shock-absorbers for cars.