Page 104 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, University Preparation
2. Investigating the Concept of the Derivative Function
Sample problem: What does the limit
f(4 + h) – f(4)
lim = 8 indicate about the h→0 h
graph of the function f(x) = x2? The graph of a general function y = f(x)?
1.6 compare, through investigation, the calcula- tion of instantaneous rates of change at a point (a, f(a)) for polynomial functions
[e.g., f(x) = x2, f(x) = x3], with and without
simplifying the expression f(a + h) – f(a) h
before substituting values of h that approach zero [e.g., for f(x) = x2 at x = 3, by determining
f(3 + 1) – f(3) = 7,
f(3 + 0.1) – f(3) = 6.1, 0.1
1
f(3 + 0.01) – f(3)
= 6.01, and
0.01
f (3 + 0.001) – f (3) = 6.001, and
0.001 f(3 + h) – f(3)
by first simplifying h
as (3+h)2 –32
h = 6 + h and then substituting
the same values of h to give the same results]
By the end of this course, students will:
2.1 determine numerically and graphically the intervals over which the instantaneous rate
of change is positive, negative, or zero for a function that is smooth over these intervals (e.g., by using graphing technology to exam- ine the table of values and the slopes of tan- gents for a function whose equation is given; by examining a given graph), and describe the behaviour of the instantaneous rate of change at and between local maxima and minima
Sample problem: Given a smooth function for which the slope of the tangent is always positive, explain how you know that the function is increasing. Give an example of such a function.
2.2 generate, through investigation using tech- nology, a table of values showing the instan- taneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative,
f ’(x) or dy , and make connections between dx
the graphs of f (x) and f ’(x) or y and dy dx
[e.g., when f (x) is linear, f ’(x) is constant; when f(x) is quadratic, f’(x) is linear; when f(x) is cubic, f’(x) is quadratic]
Sample problem: Investigate, using patterning strategies and graphing technology, relation- ships between the equation of a polynomial function of degree no higher than 3 and the equation of its derivative.
2.3
determine the derivatives of polynomial func- tions by simplifying the algebraic expression
f(x + h) – f(x)
h and then taking the limit of the
simplified expression as h approaches zero
lim f(x + h) – f(x) [i.e., determining h→0
lim
h→0
h
]
2.4
determine, through investigation using tech-
nology, the graph of the derivative f ’(x) or
dy
dx of a given sinusoidal function [i.e.,
f (x) = sin x, f (x) = cos x] (e.g., by generating
a table of values showing the instantaneous rate of change of the function for various values of x and graphing the ordered pairs; by using dynamic geometry software to verify graphi- cally that when f(x) = sin x, f ’(x) = cos x, and when f (x) = cos x, f ’(x) = – sin x; by using
a motion sensor to compare the displacement and velocity of a pendulum)
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
2.5
determine, through investigation using tech- nology, the graph of the derivative f ’(x) or
dy of a given exponential function [i.e., dx
f(x)=ax (a>0,a≠1)][e.g.,bygeneratinga table of values showing the instantaneous rate of change of the function for various values
of x and graphing the ordered pairs; by using dynamic geometry software to verify that when f(x) = ax, f ’(x) = kf(x)], and make con- nections between the graphs of f(x) and f’(x)
or y and dy [e.g., f(x) and f’(x) are both dx f ’(x)
exponential; the ratio f(x) is constant, or f ’(x) = kf (x); f ’(x) is a vertical stretch from
the x-axis of f(x)]
Sample problem: Graph, with technology, f(x)=ax (a>0,a≠1)andf’(x)onthesame set of axes for various values of a (e.g., 1.7, 2.0, 2.3, 3.0, 3.5). For each value of a,
investigate the ratio f ’(x) for various values f (x)
of x, and explain how you can use this ratio to determine the slopes of tangents to f (x).
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