unit 1 test Flashcards
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
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
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
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
lim 3
x->2
3
rationalizing technique
multiply by conjugate
conjugate (sq rt. x) + 4 = (sq rt. x) - 4
squeeze theorem
how does graph look
3 lines sandwiched together
solving for point that connects all 3 lines in the middle
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
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”
finding limits of infinity
figure out HA of f(x)
the HA is the answer
if no HA, the limit DNE
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
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
what are the functions that are continuous at any point IN THEIR DOMAIN
polynomial, rational, radical, basic, and trig functions(sin,cos, tan)
approaching from the left
-
approaching from the right to
+
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-
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
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
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
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
1 plus undefined
DNE
lim. (for) sin or cos x
x -> infinity
0
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
No
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
