Page 98 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
2. Combining Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
Sample problem: In general, does the speedo- meter of a car measure instantaneous rate of change (i.e., instantaneous speed) or average rate of change (i.e., average speed)? Describe situations in which the instantaneous speed and the average speed would be the same.
1.6 determine, through investigation using various representations of relationships (e.g., tables of values, graphs, equations), approximate instan- taneous rates of change arising from real-world applications (e.g., in the natural, physical, and social sciences) by using average rates of change and reducing the interval over which the average rate of change is determined
Sample problem: The distance, d metres, travelled by a falling object in t seconds is represented by d = 5t2. When t = 3, the instantaneous speed of the object is 30 m/s. Compare the average speeds over different time intervals starting at t = 3 with the instantaneous speed when t = 3. Use your observations to select an interval that can be used to provide a good approximation of the instantaneous speed at t = 3.
1.7 make connections, through investigation, between the slope of a secant on the graph
of a function (e.g., quadratic, exponential, sinusoidal) and the average rate of change
of the function over an interval, and between the slope of the tangent to a point on the graph of a function and the instantaneous rate of change of the function at that point
Sample problem: Use tangents to investigate the behaviour of a function when the instan- taneous rate of change is zero, positive, or negative.
1.8 determine, through investigation using a vari- ety of tools and strategies (e.g., using a table of values to calculate slopes of secants or graphing secants and measuring their slopes with technology), the approximate slope of the tangent to a given point on the graph of
a function (e.g., quadratic, exponential, sinu- soidal) by using the slopes of secants through the given point (e.g., investigating the slopes
of secants that approach the tangent at that point more and more closely), and make con- nections to average and instantaneous rates of change
1.9 solve problems involving average and instan- taneous rates of change, including problems
arising from real-world applications, by using numerical and graphical methods (e.g., by using graphing technology to graph a tangent and measure its slope)
Sample problem: The height, h metres, of a ball above the ground can be modelled by the function h(t) = – 5t2 + 20t, where t is the time in seconds. Use average speeds to determine the approximate instantaneous speed at t = 3.
By the end of this course, students will:
2.1 determine, through investigation using graph- ing technology, key features (e.g., domain, range, maximum/minimum points, number of zeros) of the graphs of functions created by adding, subtracting, multiplying, or dividing functions [e.g., f(x) = 2–x sin 4x, g(x) = x2 + 2x,
h(x) = sin x ], and describe factors that affect cos x
these properties
Sample problem: Investigate the effect of the behaviours of f(x) = sin x, f(x) = sin 2x, and f(x) = sin 4x on the shape of
f(x) = sin x + sin 2x + sin 4x.
2.2 recognize real-world applications of combi- nations of functions (e.g., the motion of a damped pendulum can be represented by a function that is the product of a trigonometric function and an exponential function; the fre- quencies of tones associated with the numbers on a telephone involve the addition of two trigonometric functions), and solve related problems graphically
Sample problem: The rate at which a conta- minant leaves a storm sewer and enters a lake depends on two factors: the concentra- tion of the contaminant in the water from the sewer and the rate at which the water leaves the sewer. Both of these factors vary with time. The concentration of the contaminant, in kilograms per cubic metre of water, is given by c(t) = t2, where t is in seconds. The rate at which water leaves the sewer, in cubic
metres per second, is given by w(t) = 1 . t4 +10 Determine the time at which the contaminant leaves the sewer and enters the lake at the maximum rate.
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