Question

I will post a graph and using it you have to figure out 4 things.
(i.) (lim x->60-) f(x)

(ii.) (lim x->60+) f(x)

(iii.) What can you conclude about (lim x->60) f(x)? How is this shown by the graph?

(iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities?

Answer 1

do u know what \(x\to60-\) means??

Answer 2

as x approaches 60

Answer 3

from the left

Answer 4

@helder_edwin

Answer 5

yes. sorry, my laptop's battery went down.

Answer 6

now put yourself a little bit to the left of x=60 and then go up until u meet the function and then follow tha graph going leftwards with passing x=60

Answer 7

will the answer be the closed dot or the open dot at x=60?

Answer 8

if u go up a llitle bit to the left of x=60 u get to y=68
right?

Answer 9

yes

Answer 10

sorry gotta go!!
remember \(x\to60-\) means x approaches 60 from the left but it never actually reaches x=60 so
\[ \large \lim_{x\to60-}f(x)=68 \]
in your graph.

Answer 11

ok so (i.) is 68 so (ii.) is 68 too?

Answer 12

recheck it (i.) is 56

Answer 13

but u r right (ii.) is 68

Answer 14

i think
1) (i) is 56 bcz it is to the left of 60
2) is 68
3) i think one sides limit exist at 60 not two sided

Answer 15

@mukushla yes ur right sorry

Answer 16

np sara :)

Answer 17

as u can see and aliza mentioned (lim x->60-) f(x) \(\neq\) (lim x->60+) f(x)

so the limit (lim x->60) f(x) does not exist.

Answer 18

but the lim x->60- and lim x->60+ do exist right? and what would the answers be to these questions?

(iii.) How is this shown by the graph?

(iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities?

Answer 19

yes the left and right hand limits exist, just not the limit itself

Answer 20

left hand limit is shown and found by approaching x=60 from the left

right hand limit is shown and found by approaching x=60 from the right

Answer 21

basically what the previous posters said

Answer 22

but (iii.) How is this shown by the graph?

(iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount at its discontinuities?

Answer 23

(iii) there is no limit but how is this shown by the graph?

Answer 24

you use the graph to find the left and right hand limits

Answer 25

you basically start at x = 60, then you go to the left a bit as described above and you approach closer and closer to 60. You'll end up at y = 56

Answer 26

to find the right hand limit, you start at x = 60, go to the right a bit, then approach it closer and closer to get y = 68

Answer 27

does that make sense?

Answer 28

yes i understand (i) and (ii) perfectly but i ned help with (iii) and (iv)

Answer 29

it's shown by the graph because you use the graph to find these limits

Answer 30

as for part iv, it looks like they increase the price every 20 miles

Answer 31

increase the price by 12 dollars every 20 miles?

Answer 32

yes, and this increase isn't continuous since it only happens at the beginning of each 20 mile interval

Answer 33

ok thanks for ur help! 2 more please?

Answer 34

sure

Answer 35

The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable income of x dollars.

T(x) = I WILL POST TABLE

(i) Determine whether T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
(iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

Answer 36

i) plug in x = 6061 into the first and second pieces (separately). If they produce the same T(x) value, then it's continuous at x = 6061

Answer 37

they produce the same T(x) value

Answer 38

so it's continuous at x = 6061

Answer 39

ok so for (ii) you plug in 32473 into the second and third pieces?

Answer 40

yes

Answer 41

and see if they produce the same T(x) value

Answer 42

and they are equal so continuous at 32473 too.

Answer 43

yes

Answer 44

so (iii) is not applicable right?

Answer 45

you are correct, it looks like T(x) has no discontinuities (but I would check all possible points of discontinuity)

Answer 46

yes it doesnt i checked. thanks! 1 more?

Answer 47

go for it

Answer 48

and also for the limit question, for (i) and (ii) helder edwin gave me a different answer so im confused now..

Answer 49

you should have the following

\[ \large \lim_{x\to60^{-}}f(x) = 56 \]

\[ \large \lim_{x\to60^{+}}f(x) = 68 \]

Answer 50

ok. im just confused because there is a open dot at 68 and closed dot at 56 at x=60

Answer 51

and the question is A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.

Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2

Answer 52

the open and closed dot are there to make sure that the function doesn't have multiple outputs for x = 60

Answer 53

Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2

Sk: 1 + 4 + 7 + . . . + (3k - 2) = k(3k - 1)/2

Answer 54

Sk: 1 + 4 + 7 + . . . + (3k - 2) = k(3k - 1)/2

Sk+1: 1 + 4 + 7 + . . . + (3(k+1) - 2) = (k+1)(3(k+1) - 1)/2

Simplify the last line