Chapter 1 - Functions and Models

Question 1

Function

Answer 1

a group of/ rule which maps each element of a set called the domain into another set called the range and has only one output.

Question 2

Euler’s notation

Answer 2

f(x)/ (f^-1)

Question 3

Trigonometric Values

Answer 3

0 0 1 0
——————–
π/6 (1/2) √3/2 ( 1/√3)
——————–
π/4 (√2/2) (√2/2) 1
——————-
π/3 (√3/2) 1/2 (1/√3)
——————-
π/2 1 0 undef.

Question 4

composition of functions

Answer 4

f(g(x))

Question 5

exponential function

Answer 5

y=ab^x

Question 6

inverse function

Answer 6

f(g(x)) = x

g(f(x)) = x

Question 7

logarithm rules

Answer 7

log b (xy) = log b X + log b Y

log b ( x/y) = log b X - log b Y

log b ( x ^p) = p log b X

Question 8

Intermediate Value Theorem

Answer 8

if f(x) is continuous on the interval (a,b) and N is between f(a) and f(b) so f(a) < N < f(b) or f(b) < N f(a). Then there exists c with a < c < b such that f(c) = N.

In geometric terms it says that if any horizontal line y = N is given between y = f(a) and y = f(b), then the graph of f can’t jump over the line. It must intersect y = N somewhere.

• useful for showing that an equation has a solution in some interval

Question 9

What functions are continous?

Answer 9

polynomials any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Sinx and cosx are continuous

Question 10

continuous on an interval

Answer 10

a function f is continuous on an interval if it is continuous at every number in the interval. (The function needs to be continuous from the right at the left endpoint of the interval and from the left at the right endpoint of the interval)
example:
- interval [-1,1]
- f(x) needs to be continuous from the right at -1 and from the left at 1.

Question 11

A function is continuous at a number if

Answer 11

lim x-> a f(x) = f(a)

Question 12

If f(x) and g(x) are continuous at a and c is a constant, then the following functions are also continuous at:

Answer 12

f + g
f -g
cf
fg
f/g if g does not equal 0

Question 13

rational function

Answer 13

is a function that can be expressed as the ratio of two polynomials. It has the form:

f(x)=P(x)/Q(x)

• they are continuous on their domain
• you can find the limit by just plugging in the x-value

Question 14

The following types of functions are continuous at every number in their domains:

Answer 14

polynomials, rational functions, root functions, trigonometric functions

Question 15

Continuity of Composite Functions Theorem

Answer 15

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) is continuous at a

Question 16

Vertical asymtope

Answer 16

the line x = a
example:
2x/x-3
the vertical asymtope is x=3

Question 17

When can’t limit laws be applied?

Answer 17

when we are dealing with infinite limits, because infinity is not a number

Question 18

The function has a vertical asymptote at x = a if…

Answer 18

lim x-> a f(x) = -/+ infinity

Question 19

Types of limits

Answer 19

lim x-> a f(x)
lim x-> a+ f(x)
lim x-> a- f(x)
lim x-> -/+ infinity f(x)
(there are more I just can’t attach refer to https://quizlet.com/1007618962/chapter-1-in-the-essential-calculus-early-transcendentals-flash-cards/)

Question 20

The two-sided limit lim x-> a f(x) exists when…

Answer 20

the one-sided limits exist and are both equal

Question 21

A function is continuous at a if…

Answer 21

1) f(a) is defined
2) lim x->a exists
3) limx->a f(x) = f(a)
(the same is true for the left and right limits)

Question 22

A function is continuous on an interval if…

Answer 22

f(x) is continuous at c for all c in the interval. (if the left endpoint a is included in I, then f(x) has to be continous from the right at a)

Question 23

Limit laws

Answer 23

lim (f±g) = lim f ± lim g
lim (c ⋅ f) = c ⋅ lim f
lim (fg) = lim f ⋅ lim g
lim (f/g) = lim f / lim g for lim g ≠ 0
lim √f(x) = √lim f(x)
lim c = c
lim xⁿ = aⁿ
lim x = a

Question 24

Squeeze Theorem

Answer 24

If f(x) ≤ g(x) ≤ h(x) for all x near a
and lim x→a f(x) = limx→a h(x) = L
then
limx→a g(x) = L
- the squeeze theorem also applies to one-sided liits and limits x -> -/+∞

Question 25

lim x -> 0 six(x)/x

Answer 25

=1

Question 26

if -f(x) = f(-x)

Answer 26

odd example: f(x) = x^5 + x

Question 27

Power functions

Answer 27

polynomial functions of the form f(x) = x^n

Question 28

Rational functions

Answer 28

Let P(x) and Q(x) be polynomials. Then R(x) = P(x) / Q(x) is a rational function. The domain is the set of real numbers such that Q(x) does not equal 0. vertical asymtopes when Q(X) = 0

Question 29

Trigonometric Functions

Answer 29

SOH CAH TOA (sine is opposite over hypotenuse, etc.)

Question 30

sinπ/3

Answer 30

√3/2

Question 30

sinπ/6

Answer 30

1/2

Question 31

sin(π)

Answer 31

0

Question 32

cosπ/6

Answer 32

√3/2

Question 33

cosπ/3

Answer 33

1/2

Question 34

cosπ/2

Answer 34

0

Question 35

(cosx, sinx)

Answer 35

(x,y)

Question 36

tanx

Answer 36

sinx/cosx -vertical asymtopes at x = -π/2, π/2, 3π/2

Question 37

If h(t) is a horizontal line then...

Answer 37

the limit does not exist because h(t) does not approach any number

Question 38

When using limit laws...

Answer 38

Assume the limits of f(x) and g(x) exist

Question 39

If f(x) < g(x) when x is approaching a and they both exist then...

Answer 39

lim x -> a f(x) < lim x -> a g(x)

Question 40

The function discontinuous if

Answer 40

- when the graph has a break example: x^2 - x - 2 / x - 2 - when the left and right limits do not agree example: greatest integer function continuities: - jump - infinite - removable

Question 41

To prove that a function is continous on an interval

Answer 41

need to prove that the function is continous on the right of the interval and continous from the left of the interval

Question 42

limits of composite functions

Answer 42

if and only if both functions are continous and the limit exists

Question 43

then the composite function of f(g(x)) is continuous at a

Answer 43

if g is continous at a and f is continuous at a example F(x) = 1/√x^2 + 7 - 4 F(x) = f(g(h(k(x))) f(x) = 1/x g(x) = x - 4 h(x) = √x. k(x) = x^2 + 7 check the domain of all of the input in order to find the domain of the function

Question 44

Limits to infinity

Answer 44

describes the end behavior of a function, usually for vertical asymtopes - like lim x -> 0 (1/x^2) = ∞

Question 45

Limits at infinity

Answer 45

The dependent variable approaches a finite number as the independent variable becomes arbitrarily large lim x→∞ f(x) = L - usually for horizontal asymtopes

Question 46

When evaluating a large fraction polynomial

Answer 46

- divide the numerator and the denominator by the highest power of x in the denominator - multiply the numerator and denominator by the positive/negative reciprocal (usually to get rid of the square root)