Advanced Calculus – Introduction to differentiation

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

• Continuous and discontinuous functions
• Derivative
• Product, quotient and chain rule
• Rate of change

Content:

MA-C1.1

• Distinguish between continuous and discontinuous functions, identifying key elements which distinguish each type of function
  • sketch graphs of functions that are continuous and compare them with graphs of functions that have discontinuities
  • describe continuity informally, and identify continuous functions from their graphs
• Describe the gradient of a secant drawn through two nearby points on the graph of a continuous function as an approximation of the gradient of the tangent to the graph at those points, which improves in accuracy as the distance between the two points decreases
• Examine and use the relationship between the angle of inclination of a line or tangent, \( \theta \), with the positive x-axis, and the gradient, \( m \), of that line or tangent, and establish that \( \tan \theta=m \)

MA-C1.2

• Describe the behaviour of a function and its tangent at a point, using language including increasing, decreasing, constant, stationary, increasing at an increasing rate
• Interpret and use the difference quotient \( \frac{f(x+h)−f(x)}{h} \) as the average rate of change of \( f(x) \) or the gradient of a chord or secant of the graph \( y=f(x) \)
• Interpret the meaning of the gradient of a function in a variety of contexts, for example on distance–time or velocity–time graphs

MA-C1.3

• Examine the behaviour of the difference quotient \( \frac{f(x+h)−f(x)}{h} \) as \( h \to 0 \) as an informal introduction to the concept of a limit
• Interpret the derivative as the gradient of the tangent to the graph of \( y=f(x) \) at a point \( x \)
• Estimate numerically the value of the derivative at a point, for simple power functions
• define the derivative \( f^\prime (x) \) from first principles, as \( \lim_{h \to 0}\frac{f(x+h)−f(x)}{h} \) and use the notation for the derivative: \( \frac{dy}{dx}=f^\prime (x)=y^\prime \) where \( y=f(x) \)
• Use first principles to find the derivative of simple polynomials, up to and including degree
• Understand the concept of the derivative as a function
• Sketch the  derivative function (or gradient function) for a given graph of a function, without the use of algebraic techniques and in a variety of contexts including motion in a straight line
  • establish that \( f^\prime(x)=0 \) at a stationary point, \( f^\prime(x)>0\) when the function is increasing and \( f^\prime(x)<0 \) when it is decreasing, to form a framework for sketching the derivative function
  • identify families of curves with the same derivative function
  • use technology to plot functions and their gradient functions
• Interpret and use the derivative at a point as the instantaneous rate of change of a function at that point
  • examine examples of variable rates of change of non-linear functions

MA-C4.1

• Use the formula \( \frac{d}{dx}(x^n)=nx^{n−1} \) for all real values of \( n \)
• Understand and use the product, quotient and chain rules to differentiate functions of the form \( f(x)g(x) \) , \( \frac{f(x)}{g(x)} \) and \( f(g(x)) \) where \( f(x) \) and \( g(x) \) are functions
  • apply the product rule: If \( h(x)=f(x)g(x) \) then \( h^\prime(x)=f(x)g^\prime(x)+f^\prime(x)g(x) \), or if \( u \) and \( v \) are both functions of \( x \) then \( \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx} \)
  • apply the quotient rule: If \( h(x)=\frac{f(x)}{g(x)} \) then \( h^\prime(x)=\frac{g(x)f^\prime(x)−f(x)g^\prime(x)}{g(x)^2} \), or if \( u \) and \( v \) are both functions of \( x \) then \( \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}−u\frac{dv}{dx}}{v^2} \)
  • apply the chain rule: If \( h(x)=f(g(x)) \) then \( h^\prime(x)=f^\prime(g(x))g^\prime(x) \), or if \( y \) is a function of \( u \) and \( u \) is a function of \( x \) then \( \frac{dy}{dx}=\frac{dy}{du} \times \frac{du}{dx} \)
• Calculate derivatives of  power functions to solve problems, including finding an instantaneous rate of change of a function in both real life and abstract situations
• Use the derivative in a variety of contexts, including finding the equation of a tangent or normal to a graph of a power function at a given point
• Determine the velocity of a particle given its displacement from a point as a function of time
• Determine the acceleration of a particle given its velocity at a point as a function of time