W2.3 Curve Sketching

Much of physics is concerned with the way in which one physical quantity depends on another.

Examples:

The mathematical modelling we do as physicists often leads to explicit expressions for they way in which the one quantity (y say) depends on the other (x say).

It is essential to be able to see through the mathematical statement of the relationship to extract as much sense from it as we can.

Sketching the relationship is an important step in this direction…

Sketch the following functions:

  1. ax+b
  2. kx2
  3. x3
  4. $\frac{1}{x}$
  5. $\frac{1}{x+1}$
  6. $\frac{1}{x-a}$
  7. $\frac{1}{x-a} + \frac{1}{x+a}$
  8. $\frac{1}{\left[ x^2+b^2\right ]^{1/2}}$

Then summarise (use your flipcharts) the basic guidelines for curve sketching.

Solution

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Solution

Some Help!

You should have something like this …

Graphic - Figure for CurveSketching activity solution

Here are some suggestions on what you should remember when you have to sketch a graph.

  • A sketch is not the same as a drawing or diagram; it doesn’t need a calculator nor even a ruler …and it doesn’t have to be pretty.
  • Always label the axes.
  • Some sketches have a natural scale set by some number or parameter in the function; if so mark it.

    [Sketches 1, 5, 6, 7, 8 are like this.]

  • Sometimes you can build up a sketch by deciding what bit of the function is important in a given range and sketching just that bit.

    [Sketch 7 is like this: it is just a sum of two bits like 6.]

  • Special cases help to pick out landmark points.

    [Think of x=0 in sketch 7; x=0 and x=b in sketch 8]

  • Symmetry helps.

    [Think of sketch 2.]