Download Introduction. Review of Power, Trigonometric, Exponential and

Lecture Notes 1: Review of Power,
Trigonometric, Exponential, Logarithmic,
Ablsolute Value and Piecewise Functions:
Basics
Instructor: Anatoliy Swishchuk
Department of Mathematics & Statistics
University of Calgary, Calgary, AB, Canada
MATH 265 L01 Winter 2017
Outline of Lecture
1. A Brief History of Calculus
2. What is a Function?
3. Power, Trigonometric, Exponential, Logarithmic Functions
4. Absolute Value and Piecewise Function
A Brief History of Calculus
Wilhelm Leibniz (July 1, 1646-November 14, 1716) and Isaac
Newton (December 25, 1642-March 20, 1726) are usually both
credited with the invention of calculus. Newton was the first to
apply calculus to general physics and Leibniz developed much of
the notation used in calculus today.
The basic insights that both Newton and Leibniz provided were
the laws of differentiation and integration, second and higher
derivatives, and the notion of an approximating polynomial series.
By Newton’s time, the fundamental theorem of calculus was
known.
A Brief History of Calculus
Wilhelm Leibniz
Isaac Newton
What is a Function?
A function y = f (x) is a rule for determining y when we are given
a value of x. The number f (x) is the value of f (x) at x and is
read "f of x".
For example, the rule y = f (x) = 2x + 1 is a function.
Any line y = mx + b is called a linear function.
Functions can be defined in various ways: by an algebraic formula
or several algebraic formulas, by a graph, or by an experimentally
determined table of values. Given a value of x, a function must
give at most one value of y. Thus, vertical lines are not functions.
What is a Function? Domain, Restrictions
The interval of x-values at which we are allowed to evaluate the
function is called the domain of the function.
The range of f is the set of all possible values of f (x) as x varies
throughout the domain.
√
Example: Domain for f (x) = x is [0, +∞),
√
range for f (x) = x is [0, +∞).
Restrictions for the Domain:
1) We cannot divide by zero, and
2) We cannot take the square root of a negative number.
Review of Power Functions : Definition, Examples
A function of the form y = xa, where a is a constant, is called a
power function.
Several cases:
i) a = n, where n is a positive integer: f (x) = xn; thus, for n = 1
- linear function, for n = 2 - parabola;
ii) a = 1/n, where n is a positive intege: f (x) = x1/n-root function;
iii) a = −1 : f (x) = x−1 = 1/x-reciprocal function (hyperbola).
Review of Trigonometric Functions
Consider a right angle triangle with Hypotenuse (hyp), Opposite
(opp) side, Adjacent (adj) side, and opposite to opp angle θ.
Then, there are six basic trigonometric functions:
opp
sin (abbreviation for Sine): sin θ = hyp
adj
cos (abbreviation for Cosine): cos θ = hyp
sin θ
tan (abbreviation for Tangent): tan θ = opp
=
adj
cos θ
1
csc (abbreviation for Cosecant): csc θ = hyp
=
opp
sin θ
1
=
sec (abbreviation for Secant): sec θ = hyp
adj
cos θ
adj
θ
cot (abbreviation for Cotangent): cot θ = opp
= cos
sin θ
Review of Trigonometric Functions (cont’d)
Angles can be measured in degrees (o) or in radians (abbreviation
is ’rad’). The angle given by a complete revolution contains 360o
degrees, which is the same as 2π rad. Therefore,
π
rad = 180o.
From here it follows that
180 o
) ≈ 57.3o
1 rad = (
π
For example, π6 = 30o,
and
2π = 120o ,
3
1o =
π
rad ≈ 0.017rad
180
π = 90o , and so on.
2
Review of Exponential Functions
An exponential function is a function of the form f (x) = ax,
where a is a positive constant.
There are three kinds of exponential functions depending on
whether a > 1, a = 1,
or
0 < a < 1.
Review of Exponential Functions (cont’d)
Main Properties:
• only defined for positive a
• always positive: ax > 0 for all x
• Exponent Rules:
x
1) axay = ax+y ; 2) (ax)y = axy = ayx = (ay )x; 3) aay = ax−y ; 4)
axbx = (ab)x.
• If a > 1, then ax → +∞ as x → +∞, and
ax → 0, as x → −∞.
Review of Inverse Functions
We need an inverse function to define the logarithmic function
as inverse to the exponential function.
An inverse is a function that serves to ’undo’ another function:
if f (x) produces y, then putting y into the inverse of f produces
the output x.
A function f (x) is called one-to-one if every element of the range
corresponds to exactly one element of the domain.
The Horizontal Line Test: A function is one-to-one if and only
if there is no horizontal line that intersects its graph more than
once. (Example: f (x) = x2 is not one-to-one function).
Review of Inverse Functions (cont’d)
Notation for the inverse function to f (x) : f −1(x).
Explanation: if f maps x to y, then f −1(x) maps y to x.
Cancellation Formulas for the Inverse Function:
f −1(f (x)) = x
and
f (f −1(x)) = x.
Guidelines for Computing Inverse Function:
1. Write down y = f (x)
2. Solve for x in terms of y
3. Switch the x0s and y 0s
4. The result is y = f −1(x)
Review of Logarithmic Functions
The logarithmic function with base a (denoted as loga x) is defined as an inverse function to the exponential function (we consider only the case a > 1) ax :
loga x = f (x) ⇔ af (x) = x.
The cancellation formulas for logarithmic function:
loga(ax) = x
(f or
every
x
in
R)
and
Properties of Logarithmic Function (x, y > 0):
•loga(xy) = loga x + loga y;
•loga( xy ) = loga x − loga y;
•loga(xn) = n loga x.
aloga x = x
(x > 0)
Review of Logarithmic Functions: The Natural Logarithmic Function (cont’d)
The logarithmic function with the base e ≈ 2.71828 is called the
natural logarithmic function and denoted as
ln x = loge x.
It is referred to as ’natural log.’
Change of Base Formula:
ln x
loga x =
ln a
Absolute Value
The absolute value of a number x is written as |x| and represents
the distance x from zero. We define it as
(
|x| =
x,
−x,
if
if
x≥0
x < 0.
Properties of |x| :
1. |x| ≥ 0;
2. |xy| = |x||y|; 3. |1/x| = 1/|x|, when x 6= 0;
4. √
| − x| = |x|; 5. |x + y| ≤ |x| + |y|- triangle inequality;
6. x2 = |x|.
Piecewise Function
We call function a piecewise function if it is defined by different
formulas in different parts of its domain.
Examples.
1.
(
f (x) =
1 − x,
x2 ,
if
if
x≤1
x > 1.
2. Absolute value function |x| is another example of piecewise
function.
Reference
1) Calculus: Early Transcendental, 2016, An Open Text, by
David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S
1/ucalgary/winter2016/math265/Guichard
− Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf
2) Optional Textbook: Essential Calculus, Early Transcendental,
2013, by J. Stewart, 2nd edition, Brooks/Cole