Week 1-2 Flashcards
What does the symbol ℂ represent?
ℂ represents the set of complex numbers.
What does the symbol ℝ represent?
ℝ represents the set of real numbers.
What does the symbol ℚ represent?
ℚ represents the set of rational numbers.
r E Q r can be expressed as r = p/q, and (P,Q) E ℕ
What does the symbol ℕ represent?
ℕ represents the set of natural numbers, which includes {1, 2, 3, 4, …}.
What is an irrational number?
An irrational number is a real number that cannot be expressed as a fraction of two integers.
What is infinity?
Infinity is a concept of “no end” and is not a number. It represents an unbounded quantity.
What is the golden equation and its equation?
(a + b)/a = b/a = ϕ
Define the limit of a function.
The limit of a function f(x) as x approaches a is the value that f(x) gets arbitrarily close to a.
What are the laws of logarithms?
- loga(xy) = loga(x) + loga(y)
- loga(x/y) = loga(x) - loga(y)
- loga(x^k) = kloga(x)
What is the natural logarithm?
The natural logarithm, denoted as loge is the logarithm to the base e, where e ≈ 2.718
What is the absolute value of a number a?
The absolute value
∣a∣ is the distance of a from 0 on the number line. It is positive for all a >= 0 and negative for a < 0.
What is the quadratic formula?
x = ( -b +/- sqrt(b^2 - 4ac))/2a
What is an open interval?
(a,b) is an open interval meaning the interval contains the elements in between a,b except the points a,b
What is a closed interval?
[a,b] is closed interval and it contains a,b
What is a function?
A function is an assignment from a set to another set. We write f: V –> W where V and W are sets we call V the domain and W the range or image.
Define the left and right limits.
We say that the left limit of f(x) is equal to L- when we take x arbitrarily close to “a” from the left, then f(x) gets arbitratily close to L-
lim f(x) = L-
x –> a-
Similarly, we define the right limit of f(x) as follows
lim f(x) = L+
x –> a+
When does limit exist?
We say that the limit of f(x) exists when x –> a if
lim f(x) = L- = lim f(x) = L+
x –> a- x –> a+
= L
Aspect one of the Squeeze theorem
Suppose the functions f(x) <= g(x) are well defined on a certain interval(except possibily at “a”) and their limits when x approaches “a” exist. Then the lim f(x)(x–>a ) <= lim g(x) (as x–>a)
Aspect two of the Squeeze theorem
Suppose the functions satisfy
f(x) <= g(x) <= h(x)
and the limits limf(x)(x–>a) = lim h(x)(x–>a) = L
then lim g(x)(x–>a) = L
Definition of continuous
Suppose the point “a” is in the domain of the function f(x) and f(x) is well defined at “a”. We say that f(x) is continuous at “a” is
lim f(x)( x–> a ) = f(a)
When is a function continuous?
Moreover, we say that f(x) is continuous at an interval I = (a,b) if f(x) is continued at all points contained in the interval I.
Are polynomials continuous functions? Why?
P(x) = a1 + a1x + a2x^2 + a3x^3 + …anx^n
The x terms when x approaches infinity approach zero. It then becomes a constant function which is continuous.
What is the form for rational functions?
f(x) = P(x)/Q(x)
where P(x) and Q(x) are polynomials
Rational functions are continuous except maybe at Q(x) = 0
Are trigonometric, inverse and exponential functions continuous?
Yes.
