Page 107 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

B. DERIVATIVES AND THEIR APPLICATIONS
OVERALL EXPECTATIONS
By the end of this course, students will:
1. make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;
2. solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.
SPECIFIC EXPECTATIONS
By the end of this course, students will:
1.1 sketch the graph of a derivative function, given the graph of a function that is continu- ous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes)
Sample problem: Investigate the effect on the graph of the derivative of applying vertical and horizontal translations to the graph of a given function.
1.2 recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function
1.3 determine algebraically the equation of the second derivative f ”(x) of a polynomial or simple rational function f(x), and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/ decreasing intervals, local maxima and
minima, points of inflection, intervals of con- cavity) and corresponding features of the graphs of its first and second derivatives
(e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or mini- mum, and the second derivative is zero)
Sample problem: Investigate, using graphing technology, connections between key proper- ties, such as increasing/decreasing intervals, local maxima and minima, points of inflection, and intervals of concavity, of the functions f(x) = 4x + 1, f(x) = x2 + 3x – 10,
f(x) = x3 + 2x2 – 3x, and
f(x) = x4 + 4x3 – 3x2 – 18x and the graphs of their first and second derivatives.
1.4 describe key features of a polynomial function, given information about its first and/or sec- ond derivatives (e.g., the graph of a deriva- tive, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible
DERIVATIVES AND THEIR APPLICATIONS
1. Connecting Graphs and Equations of Functions and Their Derivatives
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Calculus and Vectors
MCV4U