Question

Please see the attached image. I do not understand how I am going to go from f(x) to f''(x)....

Answer 1

image?

Answer 2

Wait

Answer 3

There

Answer 4

f(x) = 2 ( x -2) +1
Can you continue to find f'(x) and f"(x)?

Answer 5

i dont think we can just assume that is f, there may be other functions that satisfy those conditions

have you tried integration by parts?

Answer 6

How do I know what f is? Sorry I do not know where to start for this one. I am clearly lost for this one.

Answer 7

have you done integration by parts before?

Answer 8

its the reverse of the product rule:\[\int\limits{g'(x).f(x)}dx = f(x)g(x) - \int\limits{f'(x).g(x)}dx\]

Answer 9

Yes

Answer 10

OHHH! So you are suggesting to do integration by parts on the one with the second derivative!

Answer 11

\[\int\limits^2_0{x^2 f''(x)}dx = [x^2f'(x)]^2_0 - \int\limits_0^2{2x f'(x)}dx\]

Answer 12

Wow! It just hit me right now.

Answer 13

something like that :)

Answer 14

OMG! I need to review my integration properties. My final is tomorrow. Thank you for your help. ALSO, I have one more question if you do not mind. I think I can deal with this one now that I know I need to use integration by parts

Answer 15

sure

Answer 16

I just need to attach it, give me a minute

Answer 17

I will message you when I post it.

Answer 18

ok

Answer 19

Ok I posted it.

Answer 20

daamn what level maths is this?

Answer 21

ok dont worry the definition is included

Answer 22

This is Calculus first year.

Answer 23

ah as in a degree? im just curious :)

Answer 24

No this is first year in university ever.

Answer 25

how are you finding it? (dont worry im still looking at your problem)

Answer 26

Math is the hardest. The sciences are pretty straightforward, given you follow the instructor's warnings about the exam.

Answer 27

But this is from my opinion. To me, it looks like it would be easy for you (ie. math)

Answer 28

im fairly sure the lower limit is -5, i just have to go and get a drink, brb

Answer 29

sorry i dont quite understand, are you starting a math degree? what are you studying for? (im sorry if this is me being stupid)

Answer 30

no the lower limit is not -5

Answer 31

i havent seen any maths like this before unfortunatly

Answer 32

The big issue i am having with this problem is the n^2 in the denominator. I have never been exposed to such a case.

Answer 33

ah ok. so can you see that the interval is -5 to 23 ?

Answer 34

im not really using a standard method here as i haven't been taught anything on integration as a limit of a summation yet

Answer 35

consider the first term of that summation,

i = 1

we have the term:

\[\frac{7 \times14}{n^2}f(-5 + \frac{28}{n})\]

Answer 36

and we know that x* satisfies \[a<x*<a+ \Delta x\]

for all n

Answer 37

and\[ x^* = -5 + \frac{28}{n}\]

\[a< -5 + \frac{28}{n} < a + \frac{(b-a)}{n}\]

Answer 38

if we take n>0

then the only a which satisfies the first part of the ineq.
is a = -5

Answer 39

hmm i think we need

\[\le\] in the second part..

im sorry i cant help that much, i havent studied this yet.
i will probably cover this stuff next year at university

Answer 40

It is alright. Thanks for your help with the previous question.

Answer 41

no problem :) @experimentX @TuringTest @Foolformath can probably help

Answer 42

???

Answer 43

See attached

Answer 44

Or look at this one:

Answer 45

@satellite73

Answer 46

Any ideas?

Answer 47

Hey try this one I posted just now eigen. Do you know this one?

Answer 48

noo are these riemann sums?

i have a calculus book im self teaching but i havent got that far yet

Answer 49

still on its differentiation section

Answer 50

i can see how they are identical, i'm struggling to express it mathematically

Answer 51

its because n tends to infinity, so the first term tends to f(1) in both series, and

f(1 + (i-1)/n) tends to f(1 + i/n) for all integer i

Answer 52

(i think)

Answer 53

looks like upper and lower Riemann sums

Answer 54

show they become the same integral then?

Answer 55

any information given on the continuity of function??

Answer 56

For which one experimentX?

Answer 57

Ok let me post out the entire question. Those were only two of the choices given.

Answer 58

c)
definite integral gives the area under the curve.

Answer 59

looks like d) the lower and upper Riemann sums also give the area under the curve.

Answer 60

http://pirate.shu.edu/~wachsmut/ira/integ/riemann.html
Lemma 7.1.10: Riemann Lemma

Answer 61

How is "a" the same as "b"? I have been trying to get to what they showed in (a) but I wasn't able to. How did they get the one in (a)?

Answer 62

they will not be same ... as n tends to infinity, they will be close enough to each other.

Answer 63

So then why is it that all of the above is the answer?

Answer 64

because that is right ...
1/n will be finitely small for large value of n, similarly, f(1 + i/n) and f(1 + i-1/n) will have close value, so they will be almost same.

Answer 65

if we can determine a number epsilon > 0 such that the difference of two sums is less than epsilon, it proves that they are very close to each other.

Answer 66

Ok I understand. I will keep that in mind. There is one more question, if you are still willing to help. Its alright if you have other things to do.

Answer 67

yeah ... go on

Answer 68

Ok it is this question. Let me just attach it.

Answer 69

Pesky n^2.... >_<

Answer 70

How would I go about with this problem?

Answer 71

somehow looks like a or c, but ... there's stupid front part.

Answer 72

b also fits form the point of view of limit.

Answer 73

i/n = somehow x

Answer 74

That n^2 is throwing me off. I am not used to that. I am just used to seeing:

f(xi)*delta x format

Where delta x
= b-a/n

Answer 75

i/n must be somehow related to x, and 1/n is equivalent to dx

Answer 76

you mean the \(n^2\) in the denominator?

Answer 77

Yes

Answer 78

In the boxed: That is what I am used to seeing:

Answer 79

where delta x = (b-a)/n

Answer 80

7 is giving the trouble ...

Answer 81

i think without 7 it will be equal to b)

Answer 82

one n in the denominator is from \(\Delta x\)

Answer 83

Oo.. still wrong.

Answer 84

the other i/n is related to x

Answer 85

Then what do I do with the other n?

Answer 86

THE OTHER N IS RELATED TO X ALONG WITH i

Answer 87

man now this is bugging me.

Answer 88

If there were no 14, then b would have been correct.