Question

can someone please explain this to me for a medal

Answer 1

imagine it is a straight line between f(3) and f(4)
what is the slope of that line?
use that to make an equation for the line between x=3 and x=4, and use that equation to estimate f(3.5)

Answer 2

i got 16

Answer 3

can you show your work so i don't have to do it myself?
that way i can just look for mistakes

Answer 4

wait i did it differently idk what youre talking about lol

Answer 5

u can use any method to approximate?

Answer 6

yeah

Answer 7

i mean find the slope of the line from x=3 to x=4 with \[{f(4)-f(3)\over4-3}\]I'm not sure which way Marki has in mind

Answer 8

ok familiar with mean value theorem ?

Answer 9

oh ok

Answer 10

oh yes ,first i thought of Euler approximation then i thought naa it will complicate it only :P

Answer 11

f(4)-f(3)

Answer 12

now what

Answer 13

you need the slope
[f(4)-f(3)]/(4-3)

Answer 14

f(1)

Answer 15

4-3=1...

Answer 16

\[f(4)-f(3)\ne f(1)\]

Answer 17

use your chart

Answer 18

this is what @TuringTest doing

mean value theorem state for a continuous function f in interval a,b
there is c in (a,b)
s.t
f'(c)=f(b)-f(a)/b-a

so
f'(3.5)=f(4)-f(3)/4-3

Answer 19

4

Answer 20

no, what is f(4)?

Answer 21

8

Answer 22

oh its 6

Answer 23

oops

Answer 24

good, that is the slope
now use the point-slope form of a line to get an equation for the approximation

Answer 25

\[f(x)-f(x_1)=m(x-x_1)\]

Answer 26

naa there is no need :)

Answer 27

f'(3.5) using that formula is enough for the question

Answer 28

oooh yeah, I am rusty dang

Answer 29

ooh so i was wrong

Answer 30

but wouldn't 5 be the correct linear approximation @Marki ?

Answer 31

you got 5?

Answer 32

oh right

Answer 33

yes, but Marki disagrees

Answer 34

i got 5 also

Answer 35

yes because the interval of x=3 to x=4 is 6 wide, so f(3)+6/2 should be the linear approximation
the graph is clearly not linear, so perhaps his estimate is more accurate, but I don't see how he got it

Answer 36

i agree to 5 :O
what what im disagree with ?

Answer 37

oh, you said that all we needed was f(4)-f(3)/4-3=6

Answer 38

thank you guys(:

Answer 39

oh sorry its 6 not 5 xD

Answer 40

wait what

Answer 41

how? It should be 5 if we are using a linear approximation

Answer 42

haha well this is how MVT works xD

Answer 43

i think i agree with 5

Answer 44

how did u got 5 ?

Answer 45

I think that the mean value theorem only tells us that there *exists* a point c such that
f'(c)=f(b)-f(a)/b-a
not *that* f(c)=f(b)-f(a)/b-a
which is what you claim @Marki

Answer 46

remember this is approximation anything might work

Answer 47

the slope of the secant is 6, you agree with that right, marki?

Answer 48

yes lol , since its approximation xD
expected value for c is 4+3/2=3/5

Answer 49

yep

Answer 50

ok continue :)

Answer 51

so then we need to draw a line and use that slope to get a linear approximation