Question

Find the points on the lemniscate 2(x^2 + y^2)^2 = 25(x^2 - y^2) where the tangent is horizontal. The derivative using implicit differentiation is (-4x^3 - 4xy^2 + 25x)/(4x^3 + 4x^2 y + 25y)

Answer 1

leminscate is just the name of that particular type of curve but makes no difference

Answer 2

Where the tangent is horizontal, dy/dx=0
Just differentiate it and set the result equal to zero.

Answer 3

follow this !!
cims.nyu.edu/~kiryl/Calculus/.../Smith-implicit-differentiation.pdf

Answer 4

damn it i'd love to because my textbook doesn't actually explain it just provides examples but i've done lright so far anyway

Answer 5

but what do i do after i set it equal to zero i solve for x or y??

Answer 6

PLEASE HELP ME

Answer 7

so ok what 4(x^2 + y^2)(2x - 2yy') = 25(2x - 2yy') and then y'=0 so 4(x^2 + y^2)(2x) = 25(2x) NOW WHAT

Answer 8

it will give u a better explanation

cims.nyu.edu/~kiryl/Calculus/.../Smith-implicit-differentiation.pdf

Answer 9

yeah no i can't access it i'm now allowed oh well

Answer 10

i don't like go to nyu maybe if you wanna download it and upload it here go for it but i can't check that url

Answer 11

Find y' from that result. Set that equal to zero Then solve for x and y.

Answer 12

right well that's what i did at first and either way i was stuck there too that's why i tried this

Answer 13

You got this
4(x^2 + y^2)(2x - 2yy') = 25(2x - 2yy')

I think the 2nd factor on the left-hand side should be (2x+2y y'). So it should be
4(x^2 + y^2)(2x + 2yy') = 25(2x - 2yy')

Now solve for y'

Answer 14

you get
\[ y' = \frac{-4x^3-4xy^2+25x}{25y+4y(x^2+y^2)}=0 \]
It looks like we can solve the numerator for the x values that make it zero.

Answer 15

show me pleasee

Answer 16

Are you asking how to solve for x in the expression
\[ -4x^3-4xy^2+25x =0 \]

Answer 17

x=0 is one solution
now solve
\[ 4x^2+4y^2-25=0 \]
moving the \( 4y^2-25 \) to the right-hand side, and dividing by 4:
\[x^2= \frac{24-4y^2}{4} \]

Answer 18

Now try each of these x values in the original equation. Notice that x=0 results in an imaginary value for y, so we can discard that point.

with x^2 substituted into the original equation, solve for y, and then find x.

you should get y=±5/4

Answer 19

oh no i'd gotten 2x+2yy' for the second term where you said i got -
btw