Page 97 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
D. CHARACTERISTICS OF FUNCTIONS OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;
2. determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;
3. compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.
SPECIFIC EXPECTATIONS
1. Understanding Rates of Change
By the end of this course, students will:
1.1 gather, interpret, and describe information about real-world applications of rates of change, and recognize different ways of representing rates of change (e.g., in words, numerically, graphically, algebraically)
Sample problem: John rides his bicycle at a constant cruising speed along a flat road. He then decelerates (i.e., decreases speed) as he climbs a hill. At the top, he accelerates (i.e., increases speed) on a flat road back to his constant cruising speed, and he then acceler- ates down a hill. Finally, he comes to another hill and glides to a stop as he starts to climb. Sketch a graph of John’s speed versus time and a graph of his distance travelled versus
1.2 recognize that the rate of change for a func- time.
tion is a comparison of changes in the depen-
dent variable to changes in the independent
1.4 calculate and interpret average rates of change variable, and distinguish situations in which
of functions (e.g., linear, quadratic, exponential, the rate of change is zero, constant, or chang-
sinusoidal) arising from real-world applications ing by examining applications, including
(e.g., in the natural, physical, and social sciences), those arising from real-world situations (e.g.,
given various representations of the functions rate of change of the area of a circle as the
(e.g., tables of values, graphs, equations) radius increases, inflation rates, the rising
Sample problem: Fluorine-20 is a radioactive trend in graduation rates among Aboriginal
substance that decays over time. At time 0, youth, speed of a cruising aircraft, speed of a
the mass of a sample of the substance is 20 g. cyclist climbing a hill, infection rates)
The mass decreases to 10 g after 11 s, to 5 g Sample problem: The population of bacteria
after 22 s, and to 2.5 g after 33 s. Compare
in a sample is 250 000 at 1:00 p.m., 500 000 at
the average rate of change over the 33-s
3:00 p.m., and 1 000 000 at 5:00 p.m. Compare
interval with the average rate of change over methods used to calculate the change in
consecutive 11-s intervals.
the population and the rate of change in the
1.5 recognize examples of instantaneous rates of population between 1:00 p.m. to 5:00 p.m. Is
change arising from real-world situations, and the rate of change constant? Explain your
make connections between instantaneous reasoning.
rates of change and average rates of change
1.3 sketch a graph that represents a relationship
(e.g., an average rate of change can be used to involving rate of change, as described in
approximate an instantaneous rate of change) words, and verify with technology (e.g.,
CHARACTERISTICS OF FUNCTIONS
motion sensor) when possible
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Advanced Functions
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