Limits

Question 1

All the indeterminate forms that L’Hopital’s Rule may be able to help with are:

Answer 1

0/0 ∞/∞ 0 x ∞ 1^∞ 0^0 ∞^0 ∞ - ∞

Question 2

The conditions for L’hopital’s rule are:

Answer 2

1. Differentiable

2. The Limit Must Exist

Question 3

derivative of ln(x) is ….

Answer 3

1/x

Question 4

derivative of e^x is ….

Answer 4

e^x

Question 5

1/∞ =

Answer 5

0

Question 6

d/dx of cos(x) =

d/dx of sin(x) =

Answer 6

-sin(x)

cos(x)

Question 7

sqrt(1/x^2) =

Answer 7

1/x

Question 8

sqrt(1/x^2) * sqrt(x^2 + 4x) =

Answer 8

sqrt(1/x^2 * (x^2 + 4x))

Question 9

“The derivative of a product of two functions is the ______ times the derivative of the ______, plus the ______ times the derivative of the _______.”

Answer 9

first
second
second
first

Question 10

The conditions for a function to be continuous are:

Answer 10

1. f(a) is defined
2. the limit of the function as x approaches a exists
3. the limit of the function as x approaches a is equal to f(a)

Question 11

__________ and __________ are continuous at every point in their domains.

Answer 11

Polynomials

Rational functions

Question 12

Problem-Solving Strategy: Determining Continuity at a Point:

1. Check to see if _______. If ___________, we need go no further. The function is not continuous at a. If _____________, continue to step 2.
2. Compute ___________. In some cases, we may need to do this by first computing ____________ and _______________. If ___________ does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. If _____________, then continue to step 3.
3. Compare f(a) and_________. If ___________, then the function is not continuous at a. If ______________, then the function is continuous at a.

Answer 12

f(a) is defined f(a) is undefined f(a) is defined

the limit of the function as x approaches a
the limit of the function as x approaches a from the left
the limit of the function as x approaches a from the right
the limit of the function as x approaches a
the limit of the function as x approaches a does exist

the limit of the function as x approaches a
the limit of the function as x approaches a is not equal to f(a)
the limit of the function as x approaches a = f(a)

Question 13

e^-∞ = ___ = ______ = ______

Answer 13

e^-∞ = 1/e^∞ = 1/∞ = 0

Question 14

The answer for our limit function gives the ___________ of our graph.

Answer 14

horizontal asymptote

Question 15

The largest number of horizontal asymptotes of a function is ______________.

Answer 15

2

Question 16

The largest number of vertical asymptotes of a function is ______________.

Answer 16

there isn’t a largest number because they occur at single points.

Question 17

the function is discontinuous

Answer 17

la función es discontinua

Question 18

limx→0+ for e^1/x = _______ and limx→0− for e^1/x = ______.

Answer 18

∞ from the right

0 from the left

Question 19

The derivative of x =

Answer 19

1

Question 20

The derivative of a constant is ________

Answer 20

that constant

Question 21

The derivative of ax =

Answer 21

a

Question 22

The derivative of x2 =

Answer 22

2x

Question 23

The derivative of √x =

Answer 23

(½)x-½

Question 24

The derivative of e^x =

Answer 24

e^x

Question 25

The derivative of a^x =

Answer 25

ln(a) a^x*x’

where x’ = the derivative of x, which is 1 if the power is x.

Question 26

The derivative of ln(x) =

Answer 26

1/x

Question 27

The derivative of loga(x) =

Answer 27

1 / (x ln(a))

Question 28

The derivative of tan(x) =

Answer 28

sec^2(x)

Question 29

The derivative of sin^-1(x) =

Answer 29

1/√(1−x^2)

Question 30

The derivative of cos^-1(x) =

Answer 30

−1/√(1−x^2)

Question 31

The derivative of tan^-1(x) =

Answer 31

1/(1+x^2)

Question 32

The derivative of c*f =

Answer 32

cf’ (where c is a constant)

Question 33

The derivative of x^n

Answer 33

n*x^n−1

Question 34

The derivative of f + g =

Answer 34

f’ +g’

Question 35

The derivative of f - g =

Answer 35

f’ - g’

Question 36

The derivative of f*g =

Answer 36

f * g’ + f’ * g

Question 37

The derivative of f/g =

Answer 37

(f’ * g − g’ * f )/g^2

Question 38

The derivative of 1/f =

Answer 38

−f’/f^2

Question 39

The derivative of f( g(x) ) =

Answer 39

f’( g(x) )*g’(x)

Question 40

Ways to evaluate limits:

Answer 40

1. Just Put The Value In (substitution)
2. Factoring
3. Conjugate (The conjugate is where we change
   the sign in the middle of 2 terms)
4. Finding the overall Degree of the Function
5. L’Hôpital’s Rule
6. Proving that we can get as close as we want to the answer by making “x” close to “a” (The formal method)

Question 41

The _________ is where we change the sign in the middle of 2 terms.

Answer 41

conjugate

eg. x+2 & x-2

Question 42

By finding the _________ we can find out whether a function’s limit is ___________, or easily calculated from the coefficients.

Answer 42

overall Degree of the Function

0, Infinity, -Infinity

Question 43

It is said that the function “f” has a relative maximum value at a point c, if c belongs to (a, b), such that ____________. The maximum relative value of “f” in (a, b) is d = f (c).

Answer 43

f (c)> = f (x) for all x belonging to (a, b)

Question 44

The function “f” is said to have a relative minimum value at a point c, if c belongs to (a, b), such that _____________. The relative minimum value of f in (a, b) is d = f (c).

Answer 44

f (c) <= f (x) for all x belonging to (a, b )

Question 45

To find local maxima and minima:

If f ‘(x0 ) = 0 and there exists f’ ‘(x0 ), then:

Answer 45

f ‘’ (x0 ) > 0 => f has a relative minimum at x0.

f ‘’ (x0 ) < 0 => f has a relative maximum at x0.

Question 46

Calculation of relative maxima and minima of a function f (x) in an interval [a, b]:

1. We find ____________ and __________.
2. We carry out the ________, and calculate ____________, and if:
   f ‘’ (a) <0 we have a relative _________
   f ‘’ (a)> 0 we have a relative _________
3. It is checked whether the ____________ and the _______________ are relative maximums or minimums.
4. If there is some point of [a, b] in which the function is not differentiable, although it is continuous, we will also calculate ________________, as it could be an extreme.
5. If f is not continuous at some point x0 of [a, b], we will ________________.
6. We calculate the image (in the function) of the relative extrema.

Answer 46

the first derivative
calculate its roots

second derivative
the sign that the roots of the first derivative take on

maximum
minimum

initial point of the interval “a”
end point of the same interval “b”

the value of f at that point
study the behavior of the function in the vicinity of x0

** extrema plural for extremes

Question 47

Absolute maximum value of a function in all its domain:

f (c) is the absolute maximum value of the function f ________________ and if _______________.

Answer 47

if c belongs to the domain of f

f (c)> = f (x), for all x belonging to the domain of F

Question 48

Absolute minimum value of a function in all its domain:

f (c) is the absolute minimum value of the function f if ___________ and if _______________.

Answer 48

c belongs to the domain of f

f (c) <= f (x), for all x belonging to the domain of f

Question 49

Critical number:

If c ____________ and if _________ or _____________, then c is called the critical point of f

Answer 49

belongs to the domain of f
f ‘(c) = 0
f’ (c) does not exist

Question 50

Procedure to determine the absolute extremes of a function in the closed interval [a, b]:

1. The ____________ are obtained, and the ___________ for these numbers.
2. Find ______ and _____.
3. The largest of the values ​​found in steps 1 and 2 is the absolute _______ value, and the smallest is the absolute ______ value.

Answer 50

critical numbers of the function in (a, b)

corresponding values ​​of f are calculated

f (a) & f (b)

maximum
minimum

Question 51

Supremo de la función:
_______________.

Ínfimo de la función:
_______________.

Answer 51

Valor máximo que puede alcanzar la función

Valor mínimo que puede alcanzar la función

Question 52

The graph of y = f (x) is concave upward on those intervals where ____________.

The graph of y = f (x) is concave downward on those intervals where ____________.

If the graph of y = f (x) has a point of inflection then ________.

Answer 52

y = f “(x) > 0 (the second derivative is greater than 0)

y = f “(x) < 0 (the second derivative is less than 0)

y = f “(x) = 0 (the second derivative equals 0)

Question 53

A point where the concavity changes (from up to down or down to up) is called a ___________ (POI); note that the tangent line to a graph at a point of inflection must ____________.

Answer 53

point of inflection

cross the graph at that point

Question 54

In general, note that regardless of the sign of the slope (positive, negative or zero), the slopes of the tangent are ___________ as we move from left to right when the graph is concave down and ____________ (from left to right) when it is concave up.

Answer 54

decreasing

increasing

Question 55

The absolute value always returns a positive value, so it is impossible for _______________. At this point, we notice that an equation would have ___________.

Answer 55

the absolute value to equal a negative value

no solutions

Question 56

Given an absolute value equation, we solve it by …

Answer 56

1. Isolate the absolute value term.
2. Use |A|=B to write A=B or A=−B .
3. Solve for x .

Question 57

The absolute value function can be defined as a piecewise function:

Answer 57

f(x) = |x| = {x if x ≥ 0

-x if x < 0