Page 105 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

2.6 determine, through investigation using tech- nology, the exponential function f(x) = ax
(a > 0, a ≠ 1) for which f ’(x) = f (x) (e.g., by using graphing technology to create a slider that varies the value of a in order to deter- mine the exponential function whose graph is the same as the graph of its derivative), iden- tify the number e to be the value of a for which f’(x)=f(x)[i.e.,givenf(x)=ex,f’(x)=ex],and recognize that for the exponential function f(x) = ex the slope of the tangent at any point on the function is equal to the value of the function at that point
Sample problem: Use graphing technology to determine an approximate value of e by graph- ing f(x) = ax (a > 0, a ≠ 1) for various values
of a, comparing the slope of the tangent at a point with the value of the function at that point, and identifying the value of a for which they are equal.
2.7 recognize that the natural logarithmic func- tion f(x) = loge x, also written as f(x) = ln x, is the inverse of the exponential function f(x) = ex, and make connections between
f(x) = ln x and f(x) = ex [e.g., f(x) = ln x reverses what f(x) = ex does; their graphs are reflections of each other in the line y = x; the
composition of the two functions, eln x
or ln ex, maps x onto itself, that is, eln x = x and
ln ex = x]
2.8 verify, using technology (e.g., calculator, graphing technology), that the derivative of the exponential function f (x) = ax is
f ’(x) = ax ln a for various values of a [e.g., verifying numerically for f (x) = 2x that
f ’(x) = 2x ln 2 by using a calculator to show
(2h – 1)
that lim is ln 2 or by graphing f(x) = 2x, h→0 h
determining the value of the slope and the value of the function for specific x-values, and
comparing the ratio f ’(x) with ln 2] f (x)
Sample problem: Given f(x) = ex, verify numerically with technology using lim(ex + h – ex) that f’(x) = f(x)lne.
h→0 h
By the end of this course, students will:
3.1 verify the power rule for functions of the form f(x) = xn, where n is a natural number [e.g., by determining the equations of the derivatives of the functions f(x) = x, f(x) = x2, f(x) = x3, and f(x) = x4 algebraically using
lim f(x + h) – f(x) and graphically using slopes h→0 h
of tangents]
3.2 verify the constant, constant multiple, sum, and difference rules graphically and numeri- cally [e.g., by using the function g(x) = kf(x) and comparing the graphs of g’(x) and kf’(x); by using a table of values to verify that f’(x)+g’(x)=(f +g)’(x),givenf(x)=xand g(x) = 3x], and read and interpret proofs
involving lim f(x + h) – f(x) of the constant, h→0 h
constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required)
Sample problem: The amounts of water flow- ing into two barrels are represented by the functions f (t) and g(t). Explain what f ’(t), g’(t), f ’(t) + g’(t), and ( f + g)’(t) represent. Explain how you can use this context to veri- fythesumrule,f’(t)+g’(t)=(f +g)’(t).
3.3 determine algebraically the derivatives of polynomial functions, and use these deriva- tives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs
Sample problem: Determine algebraically the derivative of f(x) = 2x3 + 3x2 and the point(s) at which the slope of the tangent is 36.
3.4 verify that the power rule applies to functions of the form f(x) = xn, where n is a rational
number [e.g., by comparing values of the 1 slopes of tangents to the function f (x) = x 2
with values of the derivative function deter- mined using the power rule], and verify
3. Investigating the Properties of Derivatives
RATE OF CHANGE
103
Calculus and Vectors
MCV4U