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Maxima & Minima

Maths: Edexcel A Level Pure Maths, topic 7.2.2

Maxima & Minima

Calculus can be used to find the maximum and minimum values of physical quantities.

Turnings points

Turnings points

  • The turning point of a graph is any coordinate on a graph where the gradient = 0.
  • Different types of turning points include:
  • Maxima, where the function is largest.
  • Minima, where the function is the smallest.
Turning points

Turning points

  • The gradient of a graph is equal to the derivative of the graph. -The xx-coordinate of a turning point is found by setting f′(x)f\prime(x) or dydx\frac{dy}{dx} equal to zero.
  • It is called a turning point because the graph often changes direction at f'(xx) = 0.
Kinematics

Kinematics

  • We can use the same method to work out the maximum and minimum values of physical quantities if they are expressed as a function of time.
  • For example, if v=−2t2+t+4v = -2t^2+t+4:
  • dvdt=−4t+1=0→t=0.25\frac{dv}{dt} = -4t + 1 = 0 \rightarrow t = 0.25 s.
  • v=−2×0.252+0.25+4=4.38v = -2\times0.25^2 + 0.25 + 4 = 4.38 m/s (2 d.p.).
  • So there is a turning point at v=4.38v = 4.38 m/s, which is the maximum value due to the shape of the graph.

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