Question

Find the equation of the tangent line to the curve
y^3 + yx^2 + x^2 - 3y^2 = 0 at the point (0,3) @Mathematics

Answer 1

seeing how the derivative of the equation defines the slope at any given point; all you have to recall is how to write the equation of a line

Answer 2

y-Py=m(x-Px) ; where they give you P(0,3)

y-3=m(x-0) ; and m=slope, or f'

y-3=f'(0,3)(x-0) ; and m=slope, or f'

Answer 3

did you get a slope of -8/9 for this one?

Answer 4

just about to start ha

Answer 5

alright lol
let me know what you end up getting

Answer 6

k man

Answer 7

i dont get how the hint helps us for#1

Answer 8

all I did was make it look like the definition of the derivative, then said it was equal to whatever the derivative was (which I got using the chain rule) and just subbed in whatever it was telling me to

Answer 9

f(x)-f(a)
/x-a?

Answer 10

did u get 1

Answer 11

-1 for the first one

Answer 12

is what i put correct?

Answer 13

then say that is equal to f' of e^sinx-1

Answer 14

that's what I did. I used the definition you said there

Answer 15

which is e^0*-1?

Answer 16

yeah, when you evaluate the derivative at pi and that gives you -1
that's how I did it anyways

Answer 17

for b i hope u got 1/80

Answer 18

yeah that's what I got

Answer 19

k i feel better

Answer 20

hope I did it right lol

Answer 21

k so for 3:
derivative is:
3y^2*y'+y'x^2+y2x+2x-6y*y'=0

Answer 22

then i got :
y'=-2x-2xy
/3y^2+x^2-6y

Answer 23

yeah

Answer 24

really? wasnt too confident
now do i just sub in 0 and 3 for x and y

Answer 25

put it all into the equation for the tangent line and sub in your x and y

Answer 26

f(a) +f'(a)(x-a)?

Answer 27

yeah

Answer 28

to get f(a) do i have to isolate y?

Answer 29

f(a) is 3 assuming the point is on the function

Answer 30

i dont see how u could get anything other than 0 for slope

Answer 31

x is 0 and we have -2xy -2x in the numerator

Answer 32

I missed an x in the numerator of the derivative ( I factored out the 2x)

I think it's 0 then

Answer 33

ok

Answer 34

so if we did it right then L(x) = 3 then right?

Answer 35

thats what i got

Answer 36

for 4
did you get the derivative is 2x

Answer 37

you need to use logarithmic differentiation for that one

Answer 38

k

Answer 39

y(2xlnx+x)

Answer 40

yeah that's it, then I put the x^(x^2) in for the y

Answer 41

so its changing at e^-1/2
but how do i prove when <>0

Answer 42

say its increasing

Answer 43

you can either make a sketch or sub in values greater than e^-1/2 and less than e^-1/2 to check the behaviour

Answer 44

for 5 i differentiated wrt to x then wrt y
now what

Answer 45

make the dx/dt = dy/dt

Answer 46

ya so
16x(2-x)=9y(y-2)

Answer 47

how did you get that?

Answer 48

its not right but i went:
differ. wrt to x
32x*x'+18y=0

Answer 49

you need to diff wrt time

Answer 50

so 32x*dx/dt+18y*dy/dt=144

Answer 51

yeah

Answer 52

i dont know how to make the, equal

Answer 53

them

Answer 54

or does 144 become 0

Answer 55

eh