Math 150 Chapter 1 - 2.6 Flashcards Preview

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1

Definition of function

A function (or map) is a rule or correspondence that associates each element of a set
X, called the domain, with a unique element of a set Y , called the codomain.

2

x={Tom,Jerry, and Sam} and y= {3, 4, 5, 6, 7}
If the pairs are (Tom,3), (Jerry, 5) and (Sam, 7), what is the domain, range and codomain?

Domain: {Tom, Jerry and Sam}
Range : {3, 5, 7}
Codomain: {3,4,5,6,7}

3

Definition of one-to-one function

one-to-one function if it never takes on the same value twice; that is
if x1 ≠ x2, then f(x1) ≠ f(x2).

4

Definition of inverse function

Let f be one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by f^-1(y) = x f(x) = y

5

Definition of arcsine or sin^(-1)⁡

The inverse function of the sine function f(x) = sin x with the domain of -π/2 ≤ x ≤ (π)/2

6

Definition lim┬(x→a)⁡〖f(x)〗= L

the limit of f(x), as x approaches to a, equals L

7

Definition of lim┬(x→a)⁡〖f(x)〗= ∞

It means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x
sufficiently close to a, but not equal to a

8

Definition of vertical asymptote for limits

The line x = a is a vertical asymptote if at least one of the side limits equals infinity or negative infinity. (Ex. lim┬(x→a)⁡〖f(x)〗= ∞)

9

Definition of continuous (2 definitions, same meaning)

- A function f is continuous at a number a if lim┬(x→a)⁡〖f(x)〗= f(a)
- A function f is continuous from the right at a number a if lim┬(x→a^+ ) [f(x)] =f(a)
and f is continuous from the left at a if
lim┬(x→a^- ) ⁡[f(x)]=f(a)

10

Definition of discontinuous

If
(1) f is defined on an open interval containing a, except perhaps at a, and
(2) f is not continuous at a
we say that f is discontinuous at a.

11

Definition of lim┬(x→∞)⁡f(x) = L (2 things)

It means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Also this is the definition of a horizontal asymptote.

12

How do you find horizontal asymptotes?

By taking the limit as x approaches infinity and by also taking the limit as x approaches negative infinity.

13

Definition of horizontal asymptote

The line y = L is called a horizontal asymptote if lim┬(x→∞)⁡f(x) = L or lim┬(x→ -∞)⁡f(x) = L

14

What is the Factor Theorem

Let P(x) be a polynomial and r a real number. If P(r) = 0, then x-r is a factor of
P(x), i.e. P(x) = (x-r)Q(x) for some polynomial Q(x). Also, if x-r is a factor of P(x), then P(r) = 0.

15

Explain this Theorem. lim┬(x→a)⁡ f(x) ≤ lim┬(x→a)⁡ g(x)

If f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist
as x approaches a, then lim┬(x→a)⁡ f(x) ≤ lim┬(x→a)⁡ g(x)

16

What is the Squeeze Theorem?

If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and lim┬(x→a)⁡ f(x) = lim┬(x→a)⁡ h(x) = L, then
lim┬(x→a)⁡ g(x) = L

17

Explain this theorem. lim┬(x→a)⁡ f(g(x)) =
f(lim┬(x→a)⁡ g(x)) = f(b)

If If f is continuous at b and lim┬(x→a)⁡ g(x) = b then
lim┬(x→a)⁡ f(g(x)) = f(lim┬(x→a)⁡ g(x)) = f(b)

18

Explain this theorem. The composite function f o g given by (f o g)(x) = f(g(x)) is continuous at a.

If g is continuous at a and f is continuous at g(a), then the composite function f o g given
by (f o g)(x) = f(g(x)) is continuous at a.

19

What is the intermediate value theorem?

Suppose that f is continuous on the closed interval
[a,b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a, b) such that f(c) = N.

20

What is the first three limit laws? What are the conditions for the limit laws to work?

Conditions: c is a constant and both the limits lim┬(x→a) f(x) and lim┬(x→a) g(x) EXISTS.
1) lim┬(x→a)⁡[f(x)+g(x)] = lim┬(x→a) f(x)+ lim┬(x→a) g(x)
2) lim┬(x→a)⁡[f(x)-g(x)] = lim┬(x→a) f(x) - lim┬(x→a) g(x)
3) lim┬(x→a) (c(f(x))) = c[lim┬(x→a) f(x)]

21

What is the other three limit laws? What are the conditions for the limit laws to work?

Conditions: c is a constant and both the limits lim┬(x→a) f(x) and lim┬(x→a) g(x) EXISTS.
4) lim┬(x→a)⁡[f(x)g(x)] = [lim┬(x→a) f(x)] [lim┬(x→a) g(x)]
5) lim┬(x→a)⁡[f(x)÷g(x)] = [lim┬(x→a) f(x)]÷[lim┬(x→a) g(x)], if lim┬(x→a) g(x) ≠ 0
6) lim┬(x→a) [f(x)]^(p/q) = [lim┬(x→a) f(x)]^(p/q)

22

What are the two special limit laws?

1) If c is a constant, then lim┬(x→a) c = c
2 ) lim┬(x→a) x = a By DSP

23

What is Direct Substitution Property? (DSP)

If f is a polynomial or a rational function and a is in the domain of f, then lim┬(x→a) f(x) = f(a)

24

What is natural numbers denoted by and what does it represent?

The set of counting numbers (1,2,3,4,5) including 0 and is denoted by N

25

What is integers denoted by and what does it represent?

The set of natural numbers with their negatives. (-1, 0, 1) and is denoted by Z

26

What is rational numbers denoted by and what does it represent?

the set of ratios of integers and is denoted by Q

27

What is real numbers and what is it denoted by?

They are all rational numbers and numbers in between them. denoted by R

28

Definition for Logarithmic Function.

The inverse function of the exponential function f(x) = a^x is called the logarithmic function with base a

29

What are the three type of discontinuity?

Removable discontinuity
Infinite discontinuity
Jump discontinuity