unit 1 test

Question 1

HA rules

Answer 1

degree of numerator < degree of denominator: y = 0

degree of numerator = degree of denominator divide leading coefficient of numerator/ denominator

degree of numerator > degree of denominator: no HA

Question 2

how to find limits (regular) algebraically

Answer 2

plug in c into the equation
if you get 0/0 (indeterminate form), factor equation
then replug in c and that’s the answer

Question 3

find limit graphically

Answer 3

find domain(where the denominator equals 0)

holes: when roots in denominator and numerator cross out

VA:set denominator equal to 0(doesn’t include hole)

points of discontinuity: any roots in denominator

Question 4

3 types of behavior where limit DNE

Answer 4

jump discontinuity: f(x) approached different # from right side of x=c than the left side

f(x) increases or decreases w/o bond (+/- infinity)

oscillation

Question 5

lim 3

x->2

Answer 5

3

Question 6

rationalizing technique

Answer 6

multiply by conjugate

conjugate (sq rt. x) + 4 = (sq rt. x) - 4

Question 7

squeeze theorem

how does graph look

Answer 7

3 lines sandwiched together

solving for point that connects all 3 lines in the middle

Question 8

squeeze theorem

the actual formula/theorem

Answer 8

h(x) = f(x) = g(x) for all x in an open interval containing c; except possibly c itself:

lim h(x) = L = lim g(x)
x-> c               x-> c

then lim f(x) exists and is also equal to L
x-> c

Question 9

squeeze theorem

solve algebraically

find lim cos x
x-> infinity

Answer 9

find y valué limits of graph

ex. -1/x = cos x = 1/x

then find the limit of first half then second half
lim h(x)      lim g(x)
x-> c          x-> c
ex. lim -1/x  = 0 (1st half)
      x-> infinity

find limit then of f(x) if h(x) equals g(x)

then identify and state “by Squeeze Theorem”

Question 10

finding limits of infinity

Answer 10

figure out HA of f(x)

the HA is the answer

if no HA, the limit DNE

Question 11

find continuity at a point

Answer 11

function is continuous when these 3 conditions are met:

f(c) is defined

lim     f(x) exists
x->c 

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

ALL 3 conditions HAVE to be TRUE

Question 12

types of discontinuities

Answer 12

hole: removable discontinuity

the one with a hole then a defined point at the same x- value is also considered a hole

jump: jump discontinuity

Question 13

what are the functions that are continuous at any point IN THEIR DOMAIN

Answer 13

polynomial, rational, radical, basic, and trig functions(sin,cos, tan)

Question 14

approaching from the left

Answer 14

-

Question 15

approaching from the right to

Answer 15

+

Question 16

continuity on a closed interval

definition/ theorem

Answer 16

a function is continuous on the closed interval [a,b] when f is continuous on the open interval (a,b)

and

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

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

Question 17

continuity of a composite function

Answer 17

if g is continuous at c and f is continuous at g(c) then the composite function given by (f of g)(x) = f(g(x)) is continuous at c

Question 18

Intermediate Value Theorem

definition

Answer 18

if f is continuous on the closed interval [a,b], f(a) is not equal to f(b) and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k

summary: used to locate zeroes of function that are continuous in the closed interval

Question 19

intermediate value theorem

how to solve w these problems

Answer 19

[0,1]

plug in a(first number in closed bracket) into og equation
f(0)= -1

then plug in b(2nd number in closed bracket) into og equation
f(1) = 2

if a is less than 0 and his greater than 0
state “ since f(0) <0 and f(1) > 0, by the intermediate value theorem, there exists at least a number c [0,1] such that f(c) = 0

Question 20

for what value of k is the function f(x) insert piece wise function continuous at x = c

Answer 20

make it so both pereciese parts equal each other

plug in 6 and solve for k

Question 21

1 plus undefined

Answer 21

DNE

Question 22

lim. (for) sin or cos x

x -> infinity

Answer 22

0

Question 23

Theorem for infinite limits

Answer 23

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

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

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

Question 24

composite function limits

Answer 24

combines both f and g(x)
plug in if DNE: Check the limit for both sides

when moving on to second part of composite function to determine sign: check og function

if coming from below it is negative if above it is positive

Question 25

instantaneous rate of change equation

Answer 25

f(b)-f(a) / b - a

smalles difference value is best when estimating equation

Question 26

what does differentiable mean

Answer 26

if differentiated at x = c it’s continous at x=c; slope exists

Question 27

the table shows selected values of a CONTINOUS function

for 0-13

fewest possible number of times f(x) = 4

Answer 27

check y values and see how many times they switch past 4

menas that for example 3-5 is crossing 4
since function is continuous it has to cross 4

Question 28

instantaneous velocity

or velocity function

Answer 28

find derivative of position function

v(t) = s’(t)

Question 29

average velocity

function

Answer 29

f(b) - f(a)/ b-a

Question 30

position function

Answer 30

f(b) - f(a)/ b-a

Question 31

velocities are negative

Answer 31

when object is moving down or left

Question 32

position function

Answer 32

uses s(t)

Question 33

when reading longer math problems

Answer 33

circle and underline important words!

Question 34

acceleration fcuntion

Answer 34

derivative of velocity function

a(t) = v’(t)

derivative of derivative of the position function

a(t) = s’’(t)

Question 35

Can limit be determined definitively by values in table

Answer 35

No

Question 36

What is the speed of the article

Answer 36

Usually the linear function

Question 37

Squeeze thrm

Lim sin x/ x-1
x->infinity

Answer 37

-1/x = sin x/x-1 = 1/x

lim x-> infinity + = -1/infinity = 0
} lim x-> infinity sim x/x-1 = 0 by Squeeze Thm
lim x-> infinity - = -1/infinity = 0

Question 38

If asked whether something is continuous show

For example at 2

Answer 38

Show f(2)
Lim f(x)
X->2  from the right and left (positive and negative)

Question 39

Composite functions how to figure it if lim from the left or right

Answer 39

If values above, it’s positive and if values below it means negative

Or if the graph is coming from underneath it is negative and if the graph is coming from above it is positive

Question 40

Estimate f(2.25)

You are given f(3) and f(2)

Answer 40

Plug in to average rate of change type formula and solve

Question 41

Find VA and intervals where func is continuous

Answer 41

Solve for solutions (includes holes and VAs)

Then add negative infinity and positive infinity

-Use unions and parentheses for the infinity ones

Question 42

How to solve

For what values of x is h not continuous

Answer 42

Plug in values for both sides of equations

If they aren’t equal then it is not continuous

Question 43

f(x+h) - f(x)/ h

Answer 43

f’(x) is sourced from the f(x)

Question 44

If if is differentials then f’(a) is given by which formula

Answer 44

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

Question 45

Find the points where the function has horizontal tangent lines

Answer 45

Find the zeroes FIGURE IT OUTT

Question 46

Let f(2+h) -f(2)/h =5

Answer 46

F is differentiable at x=2

Because it has a slope

Question 47

If f is not continuous is f differentable?

Answer 47

No f is not differnetiable

Question 48

Inverted cones related rates problems

Answer 48

Solve for r in relation to h with similar triangles

3/h = x/r

Question 49

What is the average acceleration on the interval

Answer 49

Use velocity function to plug in to average rate of change formula

Question 50

how to find inverse coordinates

Answer 50

simply switch x and y

Question 51

how to find inverse coordinates

Answer 51

simply switch x and y

Question 52

how to find limit pf absolute value function

Answer 52

graph