Extension Calculus – Introduction to differentiation

This lesson comprises four (4) master classes focusing on:

• Rates of change with respect to time
• Exponential growth and decay
• Related rates of change

Content:

ME-C1.1

• Describe the rate of change of a physical quantity with respect to time as a derivative
  • investigate examples where the rate of change of some aspect of a given object with respect to time can be modelled using derivatives
  • use appropriate language to describe rates of change, for example ‘at rest’, ‘initially’, ‘change of direction’ and ‘increasing at an increasing rate’
• Find and interpret the derivative $\frac{dQ}{dt}$, given a function in the form $Q=f(t)$, for the amount of a physical quantity present at time $t$
• Describe the rate of change with respect to time of the displacement of a particle moving along the x-axis as a derivative $\frac{dx}{dt}$ or $\dot{x}$
• Describe the rate of change with respect to time of the velocity of a particle moving along the x-axis as a derivative $\frac{d^2x}{dt^2}$ or $\ddot{x}$

ME-C2.2

• Construct, analyse and manipulate an exponential model of the form $N(t)=Ae^{kt}$ to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation)
  • Establish the simple growth model, $\frac{dN}{dt}=kN$, where $N$ is the size of the physical quantity, $N=N(t)$ at time $t$ and $k$ is the growth constant
  • verify (by substitution) that the function $N(t)=Ae^{kt}$ satisfies the relationship $\frac{dN}{dt}=kN$, with $A$ being the initial value of $N$
  • sketch the curve $N(t)=Ae^{kt}$ for positive and negative values of $k$
  • recognise that this model states that the rate of change of a quantity varies directly with the size of the quantity at any instant
• Establish the modified exponential model, $\frac{dN}{dt}=k(N−P)$, for dealing with problems such as ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’
  • verify (by substitution) that a solution to the differential equation $\frac{dN}{dt}=k(N−P)$ is $N(t)=P+Ae^{kt}$, for an arbitrary constant $A$, and $P$ a fixed quantity, and that the solution is $N=P$ in the case when $A=0$
  • sketch the curve $N(t)=P+Ae^{kt}$ for positive and negative values of $k$
  • note that whenever $k<0$, the quantity $N$ tends to the limit $P$  as $t \to \infty$, irrespective of the initial conditions
  • recognise that this model states that the rate of change of a quantity varies directly with the difference in the size of the quantity and a fixed quantity at any instant
• solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems 

ME-C1.3

• Solve problems involving related rates of change as instances of the chain rule
• Develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied