Question

Let f be the function given by square root of x^4-16x^2
a) Find the domain of f.
b) Find f '(x)
c) Find the slope of the line normal to the graph of f at x = 5

Answer 1

a. all real numbers
b. 4x^3-32x
c. f'(5)=4(5)^3-32(5)=500-160=340, so normal slope is -1/340

Answer 2

thank you
:)

Answer 3

welcome

Answer 4

7th time
where is this question coming from?

Answer 5

domain is not all real number becauase it is square root of something, not "something"

Answer 6

but this is not the first time i have seen this exact same question
not the second time either.
would like to know where it comes from, and i can give a complete answer

Answer 7

@tserio94 please let me know where this question comes from

Answer 8

what do you mean where does it come from?

Answer 9

i mean what text, homework assignment etc? where does it come from? the answer is not all real numbers for the domain

Answer 10

yeah its a hw assignment

Answer 11

from what text?

Answer 12

or did your math teacher assign it?

Answer 13

she just assigned it

Answer 14

did she ask you to post it on open study? this is the 8th time i have seen this same question

Answer 15

yes

Answer 16

that is what i thought

Answer 17

do you know why she asked you to do that?

Answer 18

an experiment maybe?

Answer 19

i dont know why to be honest

Answer 20

is it a hight school calculus class or college?

Answer 21

highschool

Answer 22

cool
are you taking calc?

Answer 23

yes

Answer 24

nice
what this the questinon
\[\sqrt{x^4-16x^2}\]

Answer 25

to find the domain of this function?

Answer 26

i cant see it says [math processing error]

Answer 27

yeah it is not working so well tonight
square root of x^4-16x^2 maybe?

Answer 28

yeah how would you find that

Answer 29

you would have to make sure that
x^4-16x^2 was greater than or equal to zero, can you do that?

Answer 30

i think , would you have to distibute the x^4-16x^2 right?

Answer 31

no the first step would be to factor

Answer 32

can you factor this one?

Answer 33

my bad thats what i meant

Answer 34

and i got x^2(x^2-16)

Answer 35

would it be now x^2(x-4)(x+4) ?

Answer 36

yes that would work

Answer 37

now we know that x^2 is always greater than or equal to zero right?

Answer 38

yes

Answer 39

so that leaves (x-4)(x+4) do you know for which values of x this is positive?

Answer 40

i got -4 and 4

Answer 41

that is wehre it is zero, but you need to know where it is positive

Answer 42

Ah, okay, stepped out. Surprised to see the long discussion. I missed the "sqrt of" part. Sorry. Thanks s73 for catching this.

Answer 43

no problem, i am happy to help
just wondering what school this is because this is a pretty sophisticated problem for high school

Answer 44

its an ap class and idk what you mean where is it possitive

Answer 45

positive means greater than zero. you cannot take the square root of a negative number so you have to make sure the expression inside the radical is not negative

Answer 46

ap calc sounds hard

Answer 47

a. all real x with |x| > 4 (x less than -4 or greater than 4)
b. (2x^3-16x)/sqrt(x^4-16x^2)
c. f'(5) = 34/3, normal slope is -3/34

Answer 48

now one factor is x^2so you don't have to worry about it, becuase it is never negative, but (x-4)(x+4) could be negative

Answer 49

a. detail correction: should be >= (so x <= -4 or x >= 4)

Answer 50

wait so x|-4<x>4 ?

Answer 51

no

Answer 52

When I gave you (a) above I said:
a. all real x with |x| > 4 (x less than -4 or greater than 4)
but I should have said
a. all real x with |x| >= 4 (x less than or equal to -4 or greater than or equal to 4)

Answer 53

x has to be less than -4 or greater than 4
it could also be 0

Answer 54

yes, good point (no pun intended), x=0 is in the domain as well

Answer 55

but how do i write that as an answer?

Answer 56

domain = {x in R | x=0 or |x|>=4}

Answer 57

... people here seem to like "|" but ":" is better to avoid confusion, so:
domain = {x in R : x=0 or |x|>=4}

Answer 58

but what about the -4 what happened to that?

Answer 59

It's in there. If |x| >= 4 then x>=4 or x<=-4.

Answer 60

if |x|>4 that is a succinct way to write x<-4 or x > 4

Answer 61

They're equivalent.

Answer 62

what notation do they use in ap calc?

Answer 63

so its x|x=o or x>4 or x< -4

Answer 64

yes

Answer 65

do you know how to find the derivative?

Answer 66

but isnt there a shprter neater way to write that or it is good as is?
and yes , but since theres a sq root wed have to use chain rule right?

Answer 67

you can write it in simplest radical form first

Answer 68

but it is probably easiest to use the chain rule

Answer 69

ok so would it be 2x^3-16x/ sq root x^4-16^2

Answer 70

There is one nice simplification you can use, if you know this:

sgn(x) = x / |x|

It's the "signum" function and returns the sign of its argument:

sgn(x) = 0 if x=0
= 1 if x>0
= -1 if x<0

This would allow you to reduce the derivative to

2sgn(x) (x^2-8) / (x^2-16)^(1/2)

Answer 71

yeah and you can divide by 2 and also divide by x, by you have to be careful as rulnik said, because you don't know if x is positive or negative

Answer 72

so you if you simplify this way you have to take in to accound whether x is positive or negative

Answer 73

is it ok to just leave it w/the answer i got?

Answer 74

that is a good idea

Answer 75

I agree.

Answer 76

dont forget that the derivative of the square root of something should have a two in the denominator, so you should divide the numerator by 2 to get 2x^2 - 16x

Answer 77

sorry i mean 2x^3-16x

Answer 78

i am still wondering where such a difficult question came from

Answer 79

from a book or from a math teacher

Answer 80

I think that's what tserio had above, (2x^3-16x)/sqrt(x^4-16x^2) [a typo or two fixed here], so should be good to go I think.

Answer 81

I'd be curious to know if tserio's teacher catches the details as well as you two did :) Nice work.

Answer 82

i would be interested to know who it is

Answer 83

is part b good? how about the last part?

Answer 84

did your math teacher give you any other questions to ask here?

Answer 85

yes but what would be the steps needed to solve part c of this question

Answer 86

you would have to replace x by 5 to see what you get
what other questions?

Answer 87

is this an on line class or a regular one?

Answer 88

plug 5 into sq root of x^4-16x^2? and no regular class

Answer 89

Yes, (2x^3-16x)/sqrt(x^4-16x^2) with x=5 is 34/3.

Answer 90

ok so for normal you basically plug in and do reciprical?

Answer 91

Yes.

Answer 92

Negative reciprocal.

Answer 93

ok thank you
do you think you can help me with the other question i posted?

Answer 94

where?

Answer 95

Let R be the region in the xy-plane between the graphs of y = e^x and y = e^-x from x = 0 to x = 2.
a) Find the volume of the solid generated when R is revolved about the x-axis.

Answer 96

Will you join me at the other problem?