Page 103 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

A. RATE OF CHANGE OVERALL EXPECTATIONS
By the end of this course, students will:
1. demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;
2. graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative;
3. verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)
1.2 describe connections between the average rate of change of a function that is smooth (i.e., continuous with no corners) over an interval and the slope of the corresponding secant, and between the instantaneous rate of change of a smooth function at a point and the slope of the tangent at that point
Sample problem: Given the graph of f(x) shown below, explain why the instantaneous rate of change of the function cannot be determined at point P.
1.3
make connections, with or without graphing technology, between an approximate value
of the instantaneous rate of change at a given
point on the graph of a smooth function and average rates of change over intervals contain- ing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approxi- mate the slope of the tangent)
1.4
recognize, through investigation with or without technology, graphical and numerical examples of limits, and explain the reasoning
involved (e.g., the value of a function approaching an asymptote, the value of the ratio of successive terms in the Fibonacci sequence)
Sample problem: Use appropriate technology to investigate the limiting value of the terms
1 1 1 2 1 3
in the sequence (1 + 1), (1 + 2), (1 + 3), (4)
1 + 1 4, ..., and the limiting value of the series 4x1–4x13 +4x15 –4x17 +4x19 –....
1.5
make connections, for a function that is smooth over the interval a ≤ x ≤ a + h, between the average rate of change of the function over
this interval and the value of the expression
f (a + h) – f (a) , and between the instantaneous h
rate of change of the function at x = a and the
value of the limit lim f(a + h) – f(a) h→0 h
1. Investigating Instantaneous Rate of Change at a Point
y
P
3
−3
3
−3
x
RATE OF CHANGE
101
Calculus and Vectors
MCV4U