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