unit 1 Flashcards
what three conditions must be met for f(x) = L to exist?
- limf(x) from left must exist
- limf(x) from right must exist
- limf(x) from left and right must be the same value
how do we find limits on piecewise functions
- evaluate based on the domain given
- doesn’t have to approach limit from both sides
important notes regarding limits (5)
- limf(x) may be undefined if x<a>a isn’t in domain</a>
- limf(x) may not exist if left and right limits aren’t equal
- limf(x) may exist even if f(a) is undefined
- limf(x) may exist even if f(a) is defined
- limf(x) can be f(a)
limf(x) may be undefined if x<a>a isn’t in domain</a>
limit may not exist if x-values don’t approach limit from both sides due to domain
limf(x) may not exist if left and right limits aren’t equal
- limf(x) = limf(x)
- bc it must approach the same way from left nd right
limf(x) may exist even if f(a) is undefined
in case of holes
limf(x) may exist even if f(a) is defined
- in case of points that only continue one way
- i.e. root functions
limf(x) can be f(a)
graph is continuous at a
continuous function
has no breaks along its entire domain
function that has breaks along its domain is called…
discontinuous function
a function, f(x), is said to be continuous at a number, x=a, given the following conditions are met:
- f(a) must exist (get # when plugging in a)
- limf(x) must exist (must approach limit from both sides)
- limf(x) = f(a), (when u plug in a, u shld get a y-value)
types of discontinuities
- removable
- infinite
- jump
- mix of types
removable discontinuity
limits of functions w/ discontinuities (holes)
infinite discontinuity
- limits of rational functions
- limit refers to asymptotes,limit approaches infinity or -infinity
jump discontinuity
- limits of piecewise functions
- “jump” in y-vals at a limit
mix of types discontinuity
diff. types
when finding values on function given limit (looking at x-axis) rmbr to….
scan the entire y-axis!! don’t miss pts!
when drawing a function based on a limit given, rmbr to…
draw the limit as a hole
property 1
limk = k
property 2
lim x = a
property 3
lim(fx + fx) = limfx + lim fx
property 4
lim c f(x) = c (limfx)
property 5
lim(fx)(gx) = (limfx)(limgx)
property 6
lim(fx/gx) = limfx/limgx
- denom must not equal 0
property 7
limfx^n = (limfx)^n
when using properties of functions and theres a square root…
- turn sqrt into exponent of 1/2
- apply the exponent when evaluating only by turning back into sqrt for final ans
how do we format properties of functions vs substitution? when do u use each?
- properties: just write = and continue down
- subbing: change to limf(x) =
- use subbing when denom = 0
what must you do before solving any limit using properties?
- CHECK IF DENOM = 0
- IF IT DOES, STATE “DENOM=0, IDF)
- IDF = INDETERMINATE FORM
- ALSO FACTOR, CANCEL HOLES
indeterminate form
when the denominator = 0
examples of continuous functions
- polynomial
- trigonometric
- exponential
functions that are not continuous unless u define the domain
- root
- rational
if upon factoring, holes are cancelled out, can u still use properties?
- no
- instead: factor, common denominator, rationalize (conjugate), or graph
calculus is the study of
change
two fundamental problems of calculus
- area problem
- tangent problem
what’s the tangent problem
finding the slope of a line with a tangent line (one point)
new equation for slope
f(x+h)-f(x) / h
what can h not equal?
0
what does it mean when h is small? big?
small: IROC
big: AROC
why does h need to be small?
to give us an accurate tangent
steps to find AROC using new slope eqn
- find difference btwn both points and plug into h
- sub first value of interval into x in eqn
- add units/direction
how do we find the equation of a tangent line at a given x value?
- draw diagram
- find y-value at x using original eqn
- find m using slope equation tht we made
- plugged into y=mx+b to isolate for b
- final ans
derivative
a function that gives the slope of the tangent line to a function at any point
process of taking a derivative
differentiate
when you can take the derivative of a function
differentiable
