0580

Differentiation

Coordinate Geometry and Graphs

Turning a problem into a curve

  • Some questions ask for the largest or smallest value a quantity can take, and the turning point is the answer
  • The quantity must first be written as an equation in one letter only
  • Where two letters appear, the question always supplies a second fact linking them, so rearrange that and substitute
A rectangle fenced on three sides against a hatched wall, with the two equal sides labelled x and the side parallel to the wall labelled y, so a fixed length of fencing links x and y
Source: Problem-solving with differentiation by Save My Exams
  • Then differentiate, set the derivative to zero, and solve to find the value of the letter
  • Substitute that value back into the original formula, because the question asks for the quantity, not for the letter
  • Letters other than x and y are common, so a formula in A and t differentiates to dA/dt
  • To justify that the answer is a maximum, name the shape: a negative quadratic can only turn at a maximum
Exam tip

Write the derivative on its own line, then the line dy/dx = 0, then the solving. A mark is awarded simply for stating dy/dx = 0, and another for correctly solving your own quadratic, so both are available even when the derivative is wrong. Substitute the x values back into the equation of the curve, never into the derivative, to get the y coordinates.

Worked example

The largest rectangle from a fixed length

A piece of wire 36 cm long is bent to form a rectangle. Find the greatest possible area.

Solution:

  • Let one side be x cm, so the two sides of that length use 2x cm of wire
  • The remaining wire is 36 − 2x cm, shared between the other two sides, so each is 18 − x cm
  • Area A = x(18 − x), which expands to A = 18x − x²
  • Differentiate: dA/dx = 18 − 2x
  • Set it to zero: 18 − 2x = 0, so x = 9
  • Substitute back into the formula for A: A = 18 × 9 − 81 = 162 − 81
  • The greatest area is 81 cm²
  • A = 18x − x² is a negative quadratic, so its only turning point is a maximum
  • The rectangle turns out to be a 9 cm square, and its perimeter checks as 2(9 + 9) = 36