LCHL Differential Calculus

PM: Differentiation Intro/Review

GEO: Using angry birds to introduce calculus

PM: Presentation Relating Derivative to Slope

Differentiation Presentation from class

These lessons will involve the students in investigating and understanding:

Rate of change, average rate of change, instantaneous rate of change, the derivative

MIF: Get the idea - Rate of change

This will include:

 Calculus as the study of mathematically defined change, (but not necessarily change with respect to time alone - velocity can change with height, temperature can change with energy, force can change with mass, pressure can change with depth etc.)

MIF: Get the idea - Rate of change

 How to use graphs and real life examples to analyse rates of change for:

o Functions of the form f(x) = k ,where k is a constant

o Linear functions - links should be established to the slope of a line from coordinate geometry

o Functions where the rate of change varies - these will include the quadratic as well as other functions where the rate of change varies

 Instantaneous rate of change (what shows on a speedometer) as opposed to average rate of change for example over the whole course of a journey

MIF: Get the idea - Rate of change

 Equality of the instantaneous and average rates of change for linear functions

 How to find the rate of change in situations where it is not constant - need to define it at every point

o The idea of average rate of change between two points on, for example, the graph of f (x)= x² and its calculation as the slope of the line connecting the two endpoints of the interval under consideration

o That the instantaneous rate of change is not the same as the average rate of change between two points on, for example, f ( x)= x²

o That the average rate of change approaches the instantaneous rate as the interval under consideration approaches zero (the concept of a limit)

o That the instantaneous rate of change is the slope of the tangent line at the point

MIF: Slope of a Function at a Point

 The meaning of the first derivative as the instantaneous rate of change of one quantity relative to another and the use and meaning of the terms “differentiation” and notation such as dy/dx and f'(x)

 How to find the first derivatives of linear functions using the equation y = mx + c and observing the slope as the first derivative

 How to differentiate linear and quadratic functions from first principles

 Differentiation by rule of the following function types

 Polynomial

 Exponential

 Trigonometric

 Rational powers

 Inverse functions

 Logarithms

 How to find the derivatives of sums, differences, products, quotients and composition of functions of the form of the above functions

 How to use differentiation to find the slope of a tangent to a circle

 How to apply the differentiation of the above functions to solve problems

 What it means when a function is increasing/decreasing/constant in terms of the rate of change

 How to apply an understanding of the change in dy/dx from positive to zero to negative around a local maximum in order to identify a local maximum (concave downwards)

 How to apply an understanding of the change in dy/dx from negative to zero to positive

around a local minimum in order to identify a local minimum (concave upwards)

 Stationary points as points on a curve at which the tangent line has a slope of zero

 Turning points as points on a curve where the function changes from increasing to decreasing or vice versa. (Turning points are also stationary points but the converse may not be true.)

 The meaning of the second derivative as the rate of change of a rate of change at an instant

 The second derivative as being positive (first derivative is increasing) in a region where the graph of a function is concave upwards and negative (first derivative is decreasing) in a region where the graph of the function is concave downwards

 A point of inflection as a point on a curve at which the second derivative equals zero and changes sign (curve changes from concave upwards to concave downwards and the first derivative has a maximum or minimum point)

 Real life examples of the rate of change of a rate of change as in acceleration as a rate of change of velocity

 How to sketch a “slope- graph ”of a function given the graph of the function

 How to match a function with graphs of its first and second derivatives

 How to find second derivatives of linear, quadratic and cubic functions by rule

 The application of the second derivative to identify local maxima and local minima

(Students might also associate the points on a normal curve which are one standard deviation away from the mean with points of inflection referred to above.)