Unit 6: Curves and Optimization

In this unit students will:

• determine numerically and graphically the intervals over which the instantaneous rate of change is positive, negative, or zero for a function that is smooth over these intervals, and describe the behaviour of the instantaneous rate of change at and between local maxima and minima

• solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions, radical function, and other simple combinations of functions

• sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function

• recognize the second derivative as the rate of change of the rate of change, and sketch the graphs of the first and second derivatives, given the graph of a smooth function

• determine algebraically the equation of the second derivative f''(x) of a polynomial or simple rational function f(x), and make connections, through investigation using technology, between the key features of the graph of the function and corresponding features of the graphs of its first and second derivatives

• describe key features of a polynomial function, given information about its first and/or second derivatives, sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible

• sketch the graph of a polynomial function, given its equation, by using a variety of strategies to determine its key features, and verify using technology

• solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of application including those arising from real-world situations

• solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results