unit 1

Question 1

what three conditions must be met for f(x) = L to exist?

Answer 1

1. limf(x) from left must exist
2. limf(x) from right must exist
3. limf(x) from left and right must be the same value

Question 2

how do we find limits on piecewise functions

Answer 2

• evaluate based on the domain given
• doesn’t have to approach limit from both sides

Question 3

important notes regarding limits (5)

Answer 3

1. limf(x) may be undefined if x<a>a isn’t in domain</a>
2. limf(x) may not exist if left and right limits aren’t equal
3. limf(x) may exist even if f(a) is undefined
4. limf(x) may exist even if f(a) is defined
5. limf(x) can be f(a)

Question 4

limf(x) may be undefined if x<a>a isn’t in domain</a>

Answer 4

limit may not exist if x-values don’t approach limit from both sides due to domain

Question 5

limf(x) may not exist if left and right limits aren’t equal

Answer 5

• limf(x) = limf(x)
• bc it must approach the same way from left nd right

Question 6

limf(x) may exist even if f(a) is undefined

Answer 6

in case of holes

Question 7

limf(x) may exist even if f(a) is defined

Answer 7

• in case of points that only continue one way
• i.e. root functions

Question 8

limf(x) can be f(a)

Answer 8

graph is continuous at a

Question 9

continuous function

Answer 9

has no breaks along its entire domain

Question 10

function that has breaks along its domain is called…

Answer 10

discontinuous function

Question 11

a function, f(x), is said to be continuous at a number, x=a, given the following conditions are met:

Answer 11

1. f(a) must exist (get # when plugging in a)
2. limf(x) must exist (must approach limit from both sides)
3. limf(x) = f(a), (when u plug in a, u shld get a y-value)

Question 12

types of discontinuities

Answer 12

• removable
• infinite
• jump
• mix of types

Question 13

removable discontinuity

Answer 13

limits of functions w/ discontinuities (holes)

Question 14

infinite discontinuity

Answer 14

• limits of rational functions
• limit refers to asymptotes,limit approaches infinity or -infinity

Question 15

jump discontinuity

Answer 15

• limits of piecewise functions
• “jump” in y-vals at a limit

Question 16

mix of types discontinuity

Answer 16

diff. types

Question 17

when finding values on function given limit (looking at x-axis) rmbr to….

Answer 17

scan the entire y-axis!! don’t miss pts!

Question 18

when drawing a function based on a limit given, rmbr to…

Answer 18

draw the limit as a hole

Question 19

property 1

Answer 19

limk = k

Question 20

property 2

Answer 20

lim x = a

Question 21

property 3

Answer 21

lim(fx + fx) = limfx + lim fx

Question 22

property 4

Answer 22

lim c f(x) = c (limfx)

Question 23

property 5

Answer 23

lim(fx)(gx) = (limfx)(limgx)

Question 24

property 6

Answer 24

lim(fx/gx) = limfx/limgx
- denom must not equal 0

Question 25

property 7

Answer 25

limfx^n = (limfx)^n

Question 26

when using properties of functions and theres a square root...

Answer 26

- turn sqrt into exponent of 1/2 - apply the exponent when evaluating only by turning back into sqrt for final ans

Question 27

how do we format properties of functions vs substitution? when do u use each?

Answer 27

- properties: just write = and continue down - subbing: change to limf(x) = - use subbing when denom = 0

Question 28

what must you do before solving any limit using properties?

Answer 28

- CHECK IF DENOM = 0 - IF IT DOES, STATE "DENOM=0, IDF) - IDF = INDETERMINATE FORM - *ALSO FACTOR, CANCEL HOLES*

Question 29

indeterminate form

Answer 29

when the denominator = 0

Question 30

examples of continuous functions

Answer 30

- polynomial - trigonometric - exponential

Question 31

functions that are not continuous unless u define the domain

Answer 31

- root - rational

Question 32

if upon factoring, holes are cancelled out, can u still use properties?

Answer 32

- no - instead: factor, common denominator, rationalize (conjugate), or graph

Question 33

calculus is the study of

Answer 33

change

Question 34

two fundamental problems of calculus

Answer 34

- area problem - tangent problem

Question 35

what's the tangent problem

Answer 35

finding the slope of a line with a tangent line (one point)

Question 36

new equation for slope

Answer 36

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

Question 37

what can h not equal?

Answer 37

0

Question 38

what does it mean when h is small? big?

Answer 38

small: IROC big: AROC

Question 39

why does h need to be small?

Answer 39

to give us an accurate tangent

Question 40

steps to find AROC using new slope eqn

Answer 40

1. find difference btwn both points and plug into h 2. sub first value of interval into x in eqn 3. add units/direction

Question 41

how do we find the equation of a tangent line at a given x value?

Answer 41

1. draw diagram 2. find y-value at x using original eqn 3. find m using slope equation tht we made 4. plugged into y=mx+b to isolate for b 5. final ans

Question 42

derivative

Answer 42

a function that gives the slope of the tangent line to a function at any point

Question 43

process of taking a derivative

Answer 43

differentiate

Question 44

when you can take the derivative of a function

Answer 44

differentiable