Home » VCE » Maths Methods U3 & 4 Master Notes » Second Derivatives

Second Derivatives

5.7 Maximum Rates of Increase or Decrease

Maximum Rates of Increase or Decrease

  • Earlier, we noticed how the first derivative f^{\prime}(x) can be used to find maximum and minimum values.
  • Similarly, we can find maximum and minimum values for f^{\prime}(x) by using its derivative, f^{{\prime}{\prime}}(x) i.e. the second derivative.
  • Recall that f^{\prime}(x) is interpreted as the rate of increase/decrease/change, therefore f^{{\prime}{\prime}}(x) is capable of finding the maximum and minimum rate of increase or decrease.
  • A table is constructed below to illustrate the cases.
Read More »5.7 Maximum Rates of Increase or Decrease

5.5 Second Derivatives and Concavity

Second Derivatives and Point of Inflection

  • The first derivative of a function is known as f^{\prime}(x). Similarly, we denote the second derivative of a function as f^{{\prime}{\prime}}(x) or \frac{d^2y}{dx^2}.
  • If f^{{\prime}{\prime}}(x)>0 for an interval, the gradient of f(x) is increasing in the interval. The curve is said to be concave up (or have a ‘smiley face’)
Read More »5.5 Second Derivatives and Concavity