Chapter 2Elementary Functions Basic elementary functions: power, exponential, logarithmic, trigonometric, and inverse trigonometric arc-trigonometric or antitrigonometric functions.. The
Trang 11.7 S OME M ATHEMATICAL M ETHODS 17
Remark The first step, the formulation of an original hypothesis, is the most difficult part of the method
of mathematical induction This step is often omitted from the method.
1.7.3 Proof by Counterexample
A counterexample is an example which is used to prove that a statement (proposition) is
false Counterexamples play an important role in mathematics Whereas a complicated proof may be the only way to demonstrate the validity of a particular theorem, a single counter example is all that is need to refute the validity of a proposed theorem
In general, the scheme of a proof by counterexample is as follows:
1 Given a proposition: all elements a that belong to a set A also belong to a set (possess
a property) B.
2 Refutation of the proposition: one specifies an element a ∗ (counterexample) that
belongs to A but does not belong to B.
Example Proposition: Numbers in the form2 2n
+ 1, where n is a positive integer, were once thought to
be prime.
These numbers are prime for n =1, 2, 3, 4 But for n = 5, we have a counterexample, since
2 2 5 + 1 = 4294967297 = 641 × 6700417;
it is a composite number.
Conclusion: When faced with a number in the form2 2n
+1, we are not allowed to assume it is either prime
or composite, unless we know for sure for some other reason.
1.7.4 Method of Undetermined Coefficients
The method of undetermined coefficients is employed to find coefficients in expressions
(such as formulas, series expansions, solutions to mathematical equations), the form of which is either known in advance or assigned based on intuitive judgment
Example The fractional function
x+ 2
whose denominator can be rewritten in the factored form x(x +1)(x – 1), can be represented as the sum of partial fractions
A
x + B
x+ 1+
C
where A, B, C are (undetermined) coefficients whose values are to be found Equating (1.7.4.1) with (1.7.4.2), multiplying by x(x2– 1), and rearranging, one obtains
(A + B + C)x2+ (–B + C –1)x – A – 2 = 0.
For this equation to be valid for any x, the coefficients of the different powers of x must be set equal to zero.
This results in the system of linear algebraic equations
A + B + C =0, C – B –1 = 0, –A –2 = 0.
On solving this system, one determines the coefficients in (1.7.4.2):
A= –2, B= 12, C= 32.
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References for Chapter 1
Bronshtein, I N and Semendyayev, K A., Handbook of Mathematics, 4th Edition, Springer-Verlag, Berlin,
2004.
Courant, R and Robbins, H., What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd
Edition, Oxford University Press, Oxford, 1996.
Franklin, J and Daoud, A., Introduction to Proofs in Mathematics, Prentice Hall, New York, 1988 Garnier, R and Taylor, J., 100% Mathematical Proof , John Wiley & Sons, New York, 1996.
Gelbaum, B R and Olmsted, J M H., Theorems and Counterexamples in Mathematics, Springer-Verlag,
New York, 1990.
Jordan, B E and Palow, W P., Integrated Arithmetic & Algebra, Addison-Wesley, Boston, 1999.
Krantz, S G., Dictionary of Algebra, Arithmetic, and Trigonometry, CRC Press, Boca Raton, 2001.
P´olya, G., Mathematics and Plausible Reasoning, Vol 1: Induction and Analogy in Mathematics, Princeton
University Press, Princeton, 1990.
Rossi, R J., Theorems, Corollaries, Lemmas, and Methods of Proof , Wiley-Interscience, Hoboken, N.J., 2006 Thompson, J E., Arithmetic for the Practical Man, Van Nostrand Reinhold, New York, 1973.
Weisstein, E W., CRC Concise Encyclopedia of Mathematics, 2nd Edition, CRC Press, Boca Raton, 2003 Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002.
Trang 3Chapter 2
Elementary Functions
Basic elementary functions: power, exponential, logarithmic, trigonometric, and inverse
trigonometric (arc-trigonometric or antitrigonometric) functions All other elementary functions are obtained from the basic elementary functions and constants by means of the four arithmetic operations (addition, subtraction, multiplication, and division) and the operation of composition (composite functions)
The graphs and the main properties of the basic as well as some other frequently occurring elementary functions of the real variable are described below
2.1 Power, Exponential, and Logarithmic Functions
2.1.1 Power Function: y = xα (α is an Arbitrary Real Number)
2.1.1-1 Graphs of the power function
General properties of the graphs: the point (1,1) belongs to all the graphs, and y >0 for
x>0 For α >0, the graphs pass through the origin (0,0); for α <0, the graphs have the
vertical asymptote x =0(y → +∞ as x →0, x >0) For α =0, the graph is a straight line
parallel to the x-axis.
Consider more closely the following cases
Case 1: y = x2n , where n is a positive integer (n =1,2, ) This function is defined for all real x and its range consists of all y≥ 0 This function is even, nonperiodic, and
unbounded It crosses the axis Oy and is tangential to the axis Ox at the origin x =0, y =0
On the interval (–∞,0) this function decreases, and it increases on the interval (0, +∞) It
attains its minimum y =0at x =0 The graph of the function y = x2(parabola) is given in
Fig 2.1 a.
Case 2: y = x2n+1, where n is a positive integer This function is defined on the entire
x -axis and its range coincides with the y-axis This function is odd, nonperiodic, and unbounded It crosses the x-axis and the y-axis at the origin x =0, y =0 It is an increasing function on the entire real axis with no points of extremum, the origin being its inflection
point The graph of the function y = x3(cubic parabola) is shown in Fig 2.1 a.
Case 3: y = x–2n , where n is a positive integer This function is defined for all x≠ 0,
and its range is the semiaxis y >0 It is an even, nonperiodic, unbounded function having
no intersection with the coordinate axes It increases on the interval (–∞,0), decreases on the interval (0, +∞), and has no points of extremum This function has a vertical asymptote
x=0 The graph of the function y = x– 2is given in Fig 2.1 b.
Case 4: y = x–2n+1, where n is a positive integer This function is defined for all x≠ 0,
and its range is the entire y-axis It is an odd, nonperiodic, unbounded function with no
intersections with the coordinate axes This is a decreasing function on the entire real axis
with no points of extremum It has a vertical asymptote x =0 The graph of the function
y = x– 1is given in Fig 2.1 b.
19
Trang 420 ELEMENTARYFUNCTIONS
x
y=x
y=x
y=x
y=x
2
2
1 3
x
Figure 2.1 Graphs of the power function y = x n , where n is an integer.
Case 5: y = x α with a noninteger α >0 This function is defined for all* x≥ 0and
its range is the semiaxis y≥ 0 This function is neither odd nor even and it is nonperiodic
and unbounded It crosses the axes Ox and Oy at the origin x =0, y = 0 and increases
everywhere in its domain, taking its smallest value at the limit point x =0, y =0 The graph
of the function y = x1 2is given in Fig 2.2
1 2 3 4
x
y
y=x1/2 y=x1/2
Figure 2.2 Graphs of the power function y = x α , where α is a noninteger.
Case 6: y = x α with a noninteger α <0 This function is defined for all x≥ 0and its
range is the semiaxis y ≥ 0 This function is neither odd nor even, it is nonperiodic and unbounded, and it has no intersections with the coordinate axes, which coincide with its horizontal and vertical asymptotes This function is decreasing on its entire domain and has
no points of extremum The graph of the function y = x–1 2is given in Fig 2.2
* In fact, the power function y = x1/nwith an odd integer n is defined and negative for all x <0 Here,
however, it is always assumed that x≥ 0 A similar assumption is made with regard to the functions of the
form y = x m/n , where m is a positive integer and m/n is an irreducible fraction.
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2.1.1-2 Properties of the power function
Basic properties of the power function:
x α x β = x α+β, (x1x2)α = x α
1x α2, (x α)β = x αβ,
for any α and β, where x >0, x1>0, x2 >0
Differentiation and integration formulas:
(x α) = αx α–1,
x α dx=
x α+1
α+1 + C if α≠–1,
ln|x|+ C if α = –1 The Taylor series expansion in a neighborhood of an arbitrary point:
x α=∞
n=0
C n
α x α–n0 (x – x0)n for |x – x0|<|x0|,
where C α n= α (α –1) (α – n +1)
n! are binomial coefficients.
2.1.2 Exponential Function: y = ax (a > 0 , a ≠ 1 )
2.1.2-1 Graphs of the exponential function
This function is defined for all x and its range is the semiaxis y >0 This function is neither
odd nor even, it is nonperiodic and unbounded, and it crosses the axis Oy at y =1 and
does not cross the axis Ox For a >1, it is an increasing function on the entire real axis; for0 < a <1, it is a decreasing function This function has no extremal points; the axis
Oxis its horizontal asymptote The graphs of these functions have the following common property: they pass through the point (0,1) The graph of y = a x is symmetrical to the
graph of y = (1/a)x with respect to the y-axis For a >1, the function a xgrows faster than
any power of x as x → +∞, and it decays faster than any power of1/x as x → –∞ The
graphs of the functions y =2x and y = (1/2)xare given in Fig 2.3
O
1 2 3 4
x
y = 2 y = 2
y
1 2
Figure 2.3 Graphs of the exponential function.
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2.1.2-2 Properties of the exponential function
Basic properties of the exponential function:
a x1 a x2 = a x1+x2, a x b x = (ab) x, (a x1)x2 = a x1x2.
Number e, base of natural (Napierian) logarithms, and the function e x:
e= lim
n→∞
n
n
=2.718281 . , e x = lim
n→∞
n
n
The formula for passing from an arbitrary base a to the base e of natural logarithms:
a x = e x ln a.
The inequality
a x1 > a x2 ⇐⇒ x x1> x2 if a >1,
1< x2 if 0< a <1
The limit relations for any a >1and b >0:
lim
x→+∞
a x
x→–∞ a x|x|b =0 Differentiation and integration formulas:
(e x) = e x,
e x dx = e x + C;
(a x) = a x ln a,
a x dx= a x
ln a + C.
The expansion in power series:
e x =1+ x
1! +
x2
2! +
x3
3! +· · · + x n
n! +· · · =∞
k=0
x k
k!.
2.1.3 Logarithmic Function: y = logax (a > 0 , a ≠ 1 )
2.1.3-1 Graphs of the logarithmic function
This function is defined for all x >0 and its range is the entire y-axis The function is neither odd nor even; it is nonperiodic and unbounded; it crosses the axis Ox at x =1and
does not cross the axis Oy For a >1, this function is increasing, and for 0< a < 1, it is
a decreasing function; it has no extremal points, and the axis Oy is its vertical asymptote.
The common property of the graphs of such functions is that they all pass through the point (1,0) The graph of the function y = log a x is symmetric to that of y = log1/a xwith respect
to the x-axis The modulus of the logarithmic function tends to infinity slower than any power of x as x → +∞; and it tends to zero slower than any power of1/x as x → +0 The
graphs of the functions y = log2x and y = log1 2xare shown in Fig 2.4
Trang 72.1 P OWER , E XPONENTIAL , AND L OGARITHMIC F UNCTIONS 23
x
y= log x
y= log x
2
1/2
y
1 2
1 2
Figure 2.4 Graphs of the logarithmic function.
2.1.3-2 Properties of the logarithmic function
By definition, the logarithmic function is the inverse of the exponential function The following equivalence relation holds:
y= loga x ⇐⇒ x = a y,
where a >0, a≠ 1
Basic properties of the logarithmic function:
aloga x = x, log
a (x1x2) = loga x1+ loga x2,
loga (x k ) = k log a x, loga x= logb x
logb a,
where x >0, x1 >0, x2>0, a >0, a≠ 1, b >0, b≠ 1
The simplest inequality:
loga x1> loga x2 ⇐⇒ x x1> x2 if a >1,
1< x2 if 0< a <1
For any b >0, the following limit relations hold:
lim
x→+∞
loga x
x b =0, lim
x→+0x
blog
a x=0
The logarithmic function with the base e (base of natural logarithms, Napierian base)
is denoted by
loge x = ln x, where e = lim
n→∞
n
n
=2.718281 .
Formulas for passing from an arbitrary base a to the Napierian base e:
loga x= ln x
ln a.
Differentiation and integration formulas:
(ln x) = 1
x,
ln x dx = x ln x – x + C.
Expansion in power series:
ln(1+ x) = x – x
2
x3
3 –· · · + (–1)n–1
x n
n +· · · =∞
k=1
(–1)k–1x
k
k , –1< x <1