unit 1 test Flashcards by Kia Thomas

HA rules

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

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how to find limits (regular) algebraically

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

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find limit graphically

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

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3 types of behavior where limit DNE

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

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lim 3

x->2

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rationalizing technique

multiply by conjugate

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

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squeeze theorem

how does graph look

3 lines sandwiched together

solving for point that connects all 3 lines in the middle

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squeeze theorem

the actual formula/theorem

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

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squeeze theorem

solve algebraically

find lim cos x
x-> infinity

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”

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finding limits of infinity

figure out HA of f(x)

the HA is the answer

if no HA, the limit DNE

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find continuity at a point

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

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types of discontinuities

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

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

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

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approaching from the left

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approaching from the right to

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continuity on a closed interval

definition/ theorem

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-

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continuity of a composite function

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

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Intermediate Value Theorem

definition

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

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intermediate value theorem

how to solve w these problems

[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

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

make it so both pereciese parts equal each other

plug in 6 and solve for k

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1 plus undefined

DNE

lim. (for) sin or cos x

x -> infinity

Theorem for infinite limits

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

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

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

composite function limits

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

instantaneous rate of change equation

f(b)-f(a) / b - a smalles difference value is best when estimating equation

what does differentiable mean

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

the table shows selected values of a CONTINOUS function for 0-13 fewest possible number of times f(x) = 4

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

instantaneous velocity or velocity function

find derivative of position function v(t) = s’(t)

average velocity function

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

position function

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

velocities are negative

when object is moving down or left

position function

uses s(t)

when reading longer math problems

circle and underline important words!

acceleration fcuntion

derivative of velocity function a(t) = v’(t) derivative of derivative of the position function a(t) = s’’(t)

Can limit be determined definitively by values in table

What is the speed of the article

Usually the linear function

Squeeze thrm Lim sin x/ x-1 x->infinity

-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

If asked whether something is continuous show For example at 2

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

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

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

Estimate f(2.25) You are given f(3) and f(2)

Plug in to average rate of change type formula and solve

Find VA and intervals where func is continuous

Solve for solutions (includes holes and VAs) Then add negative infinity and positive infinity -Use unions and parentheses for the infinity ones

How to solve | For what values of x is h not continuous

Plug in values for both sides of equations | If they aren’t equal then it is not continuous

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

f’(x) is sourced from the f(x)

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

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

Find the points where the function has horizontal tangent lines

Find the zeroes FIGURE IT OUTT

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

F is differentiable at x=2 | Because it has a slope

If f is not continuous is f differentable?

No f is not differnetiable

Inverted cones related rates problems

Solve for r in relation to h with similar triangles 3/h = x/r

What is the average acceleration on the interval

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

how to find inverse coordinates

simply switch x and y

how to find inverse coordinates

simply switch x and y

how to find limit pf absolute value function

graph