Question

when does the slope of a tangent line become vertical?

Answer 1

for example if \(f(x)=\sqrt[3]{x}\) then the slope of the tangent line at \((0,0)\) undefined, meaning the "vertical" i guess

Answer 2

it doesn't really make sense to say "the slope of the tangent line is vertical" but i assume you mean the slope grows without bound, i.e. the tangent line itself is vertical

Answer 3

yeah it is an online question and that didn't make sense to me either

Answer 4

but now that makes more sense

Answer 5

i have dy/dx=1/(4-3y^2)

Answer 6

you wrote
\[y=\frac{1}{4-3y^2}\] did you really mean to put only \(y\)'s here?

Answer 7

no i didn't. it was a typo

Answer 8

i have to say that doesn't make sense to me
you have a derivative wrt \(x\) but only \(y\)'s in your answer

Answer 9

implicits

Answer 10

so now i am totally lost
is it
\[y=\frac{1}{4-3x^2}\] or
\[y'=\frac{1}{4-3x^2}\] or what? you cannot differentiate
\[y=\frac{1}{4-3y^2}\] wrt \(x\) because there is no \(x\) in it

Answer 11

unless somehow you are assuming that \(y\) itself is some unidentified function of \(x\)

Answer 12

\[x+y^3=4y \]=> \[dy/dx=1/(4-3y^2) \]

Answer 13

hmm hold on

Answer 14

i know that part is correct

Answer 15

yes it is i just had a moment to do the algebra

Answer 16

so i guess it is vertical when \(y=\pm\frac{2}{\sqrt{3}}\)

Answer 17

what would it's x value be it didn't accept my answer

Answer 18

just +

Answer 19

no idea, but i know how to find it
replace \(y\) by that number and see what you get
or maybe it wanted \(\frac{2\sqrt{3}}{3}\) who knows?

Answer 20

yuck the \(x\) will be ugly i think

Answer 21

probably have to plug it in the original.

Answer 22

a quick calculation gives me \(x=\frac{16}{3\sqrt{3}}\) but i would not bet money on it

Answer 23

thats what i got and it says it is wrong

Answer 24

i bet my money :(

Answer 25

but i have a little more

Answer 26

\[x=4y-y^3\]
\[x=\frac{8}{\sqrt{3}}-(\frac{2}{\sqrt{3}})^3\]\[x=\frac{8}{\sqrt{3}}-\frac{8}{3\sqrt{3}}\] \[x=\frac{16}{3\sqrt{3}}\] was my computation

Answer 27

do you know that it is right?

Answer 28

i was putting in the "smaller x-value" input. oops! it worked this time.

Answer 29

Thank you!!!