Page 108 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
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y = g(x)
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2. Solving Problems Using Mathematical Models and Derivatives
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
Sample problem: The following is the graph of the function g(x).
y
x
If g(x) is the derivative of f (x), and f (0) = 0, sketch the graph of f(x). If you are now given the function equation g(x) = (x – 1)(x – 3), determine the equation of f”(x) and describe some features of the equation of f(x). How would f(x) change graphically and alge- braically if f (0) = 2?
1.5 sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/ decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology
By the end of this course, students will:
2.1 make connections between the concept of motion (i.e., displacement, velocity, accelera- tion) and the concept of the derivative in a variety of ways (e.g., verbally, numerically, graphically, algebraically)
Sample problem: Generate a displacement– time graph by walking in front of a motion sensor connected to a graphing calculator.
Use your knowledge of derivatives to sketch the velocity–time and acceleration–time graphs. Verify the sketches by displaying the graphs on the graphing calculator.
2.2 make connections between the graphical or algebraic representations of derivatives and real-world applications (e.g., population and rates of population change, prices and infla- tion rates, volume and rates of flow, height and growth rates)
Sample problem: Given a graph of prices over time, identify the periods of inflation and deflation, and the time at which the maximum rate of inflation occurred. Explain how derivatives helped solve the problem.
2.3 solve problems, using the derivative, that involve instantaneous rates of change, includ- ing problems arising from real-world applica- tions (e.g., population growth, radioactive decay, temperature changes, hours of day- light, heights of tides), given the equation
of a function*
Sample problem: The size of a population of butterflies is given by the function
P(t) = 6000 where t is the time in days. 1 + 49(0.6)t
Determine the rate of growth in the popula- tion after 5 days using the derivative, and verify graphically using technology.
2.4 solve optimization problems involving poly- nomial, simple rational, and exponential func- tions drawn from a variety of applications, including those arising from real-world situations
Sample problem: The number of bus riders from the suburbs to downtown per day is represented by 1200(1.15)–x, where x is the fare in dollars. What fare will maximize the total revenue?
2.5 solve problems arising from real-world appli- cations by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathemati- cal results, and interpret and communicate the results
Sample problem: A bird is foraging for berries. If it stays too long in any one patch it will be spending valuable foraging time looking for the hidden berries, but when it leaves it will have to spend time finding another patch. A model for the net amount of food energy in
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*The emphasis of this expectation is on the application of the derivative rules and not on the simplification of resulting complex algebraic expressions.