Page 65 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Sample problem: Explain in a variety of ways how you can distinguish the exponen- tial function f(x) = 2x from the quadratic function f(x) = x2 and the linear function
f (x) = 2x.
By the end of this course, students will:
2.1 collect data that can be modelled as an exponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; meas- urement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph
the data
Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction.
2.2 identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)
2.3 solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations
Sample problem: The temperature of a cooling liquid over time can be modelled by x
the exponential function T(x) = 60(12 )30 + 20,
where T(x) is the temperature, in degrees Celsius, and x is the elapsed time, in minutes. Graph the function and determine how long it takes for the temperature to reach 28oC.
By the end of this course, students will:
3.1 compare, using a table of values and graphs, the simple and compound interest earned for a given principal (i.e., investment) and a fixed interest rate over time
Sample problem: Compare, using tables of values and graphs, the amounts after each of the first five years for a $1000 investment at 5% simple interest per annum and a $1000 investment at 5% interest per annum, com- pounded annually.
3.2 solve problems, using a scientific calculator, that involve the calculation of the amount, A (also referred to as future value, FV ), and the principal, P (also referred to as present value, PV ), using the compound interest formula in theformA=P(1+i)n [orFV=PV(1+i)n]
Sample problem: Calculate the amount if $1000 is invested for three years at 6% per annum, compounded quarterly.
3.3 determine, through investigation (e.g., using spreadsheets and graphs), that compound interest is an example of exponential
growth [e.g., the formulas for compound interest, A = P(1 + i)n, and present value, PV = A(1 + i )– n , are exponential functions, where the number of compounding periods, n, varies]
Sample problem: Describe an investment that could be represented by the function f(x)=500(1.01)x.
3.4 solve problems, using a TVM Solver on a graphing calculator or on a website, that involve the calculation of the interest rate per compounding period, i, or the number of compounding periods, n, in the compound interest formula A = P (1 + i ) n [orFV=PV(1+i)n]
Sample problem: Use the TVM Solver in a graphing calculator to determine the time it takes to double an investment in an account that pays interest of 4% per annum, com- pounded semi-annually.
3.5 explain the meaning of the term annuity, through investigation of numeric and graphical representations using technology
EXPONENTIAL FUNCTIONS
3. Solving Financial Problems Involving Exponential Functions
2. Solving Problems Involving Exponential Functions
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Functions and Applications
MCF3M