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AP Calculus AB > Unit 1 - Limits & Continuity > Flashcards

Unit 1 - Limits & Continuity Flashcards

1
Q

What is the secant line?

A

Straight line that connects two points on a function; equivalent to the AVERAGE ROC

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2
Q

What is the tangent line?

A

Straight line that connects two very close points or one point on a function; equivalent to the INSTANTANEOUS ROC

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3
Q

How do you read a limit?

A

“The limit x approaches (x-value) of f(x) is …”

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4
Q

T or F: Limit and y-value could be the same value but does not have to be the same value.

A

True

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5
Q

What information does a limit give you?

A

A limit tells you what point a function is APPROACHING , but it doesn’t tell you what the VALUE of that particular point is.

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6
Q

What is a one-sided limit?

A

A limit in which you define the y-value of a function as it approaches a given x-value from either the left or right side

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7
Q

How do you notate a limit from the left side?

A

Add a NEGATIVE sign after the x-value

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8
Q

How do you notate a limit from the right side?

A

Add a POSITIVE sign after the x-value

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9
Q

How can you determine what value a limit is approaching from the table?

A

As the x-values get increasingly closer to the point of interest, the y-value will also increasingly approach a value.

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10
Q

How do you evaluate a limit at a point?

A

direct substitution or factor and cancel

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11
Q

If two sides of a limit does not meet, then ….

A

The limit DOES NOT exist

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12
Q 
lim   sin(x)/x = 
x->0
A

1

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13
Q

lim (1 - cos(x))/x =

x->0

A

0

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14
Q

If you completely factor the numerator and denominator of an expression and no factors cancel, then…

A

the limit DOES NOT exist

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15
Q

When an indeterminate form appears, what strategies can you use to determine the limit

A

Complex fractions

Rationalizing with Radicals

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16
Q

Squeeze Theorem

A 
1- If g(x) <= f(x) <= h(x)
2- If lim g(x) = L and lim h(x) = L
       x->a                 x->a
Then lim f(x) = L
         x->a
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17
Q

How do you solve absolute value limits?

A

Find numbers that are close to x to substitute as x

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18
Q

Three conditions of continuity

A 
1- f(c) is defined (with c=domain)
2- lim f(x) exists
    x->c
3- lim f(x) = f(c)
    x->c
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19
Q

Three types of discontinuities

A

Hole/ Removable
Vertical Asymptote/ Non-removable
Jump/ Non-removable

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20
Q

In a function, discontinuities are found when …

A

the denominator is equivalent to 0

21
Q

If the factors in an expression cancels, then the discontinuity can be classified as…

A

hole discontinuity

22
Q

If the factors in an expression does not cancels, then the discontinuity can be classified as…

A

an vertical asymptote

23
Q

If the left side of a limit is different from the right side of a limit, then the discontinuity can be classified as …

A

jump discontinuity

24
Q

What are the steps to simplify the limit with complex functions?

A

1- Simplify the side of the function with complex fractions by multiplying by common factors
2- Re-write original equation to cancel any holes and simplify

25
Q

What are the 3 main restrictions in a function’s domain?

A

1- Denominators CAN NOT EQUAL ZERO
2- All even roots have a domain GREATER or EQUAL to 0
3- All logarithms (inverse of exponential) have a domain GREATER than 0

26
Q

Limit concept allows us to define _____________ ROC in terms of _______ ROC.

A

instantaneous

average

27
Q

Because average ROC divides change in one variable by change in other, average ROC is _________ at a point where change in the independent variable would be ____

A

undefined

zero

28
Q

A limit can be expressed in what three ways?

A

Graphically
Numerically
Analytically

29
Q

What are 3 ways that a limit might not exist at particular values of x?

A

Unbounded
Oscillating near this value
Limit from left side does not equate to that from the right side

30
Q

lim 1/x^2 =

x->0

A

Infinity

31
Q

lim lxl / x=

x->0

A

does not exist

32
Q 
lim  sin(1/x) =
x->0
A

does not exist

33
Q

lim 1/x =

x->0

A

does not exist

34
Q

What are the three ways to algebraically manipulate a limit?

A

1- cancelling out COMMON FACTORS of rational functions
2- multiplying by expression involving CONJUGATE of a sum or difference to simplify functions involving RADICALS
3- using ALTERNATE forms of TRIG functions

35
Q

A function is _________ on an interval if function is continuous at ____ point in the interval

A

continous

each

36
Q

In order for a _________-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on on side must _____ the value on the other side as well as the _____ of the function at the boundary.

A

piecewise
equal
value

37
Q

It is possible to remove a discontinuity by ________ the value of the function at that point so it _____ the value of the _____ of the function as x approaches that point.

A

defining
equals
limit

38
Q

A vertical asymptote that approaches positive infinity can be identified if ….

A

The change in the numerator is much greater than the change in the denominator in the positive direction

39
Q

A vertical asymptote that approaches negative infinity can be identified if ….

A

The change in the numerator is much greater than the change in the denominator in the negative direction

40
Q

Limits of horizontal asymptotes at infinity describes …

A

end behavior

41
Q

Horizontal asymptote is the __________ line a function __________ as it heads towards ________ (positive or negative)

A

horizontal
approaches
infinity

42
Q

If the denominator grows faster than the numerator in a function, then the function will equal …

A

0 or horizontal asympote

43
Q

If the denominator and numerator grows at about the same rate, then the function will equal ….

A

1

44
Q

If the numerator grows faster than the denominator, then the function will equal…

A

infinity

45
Q

Rank in order of least to greatest growth.
LOGS
EXPONENTIALS
POLYNOMIALS

A

Logs
Polynomials
Exponentials

46
Q

When a limit evaluates a function at infinity,

A

you can use direct substitution to evaluate

47
Q

If f is a __________ function from a to b, then _____ value between f(a) and f(b) _____ at some point in the interval [a,b].

A

continuous
every
exist

48
Q

lim x/sin(x) =

x->0

A

1

49
Q

lim x/(1 - cos(x)) =

x->0

A

0

AP Calculus AB (10 decks)

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