implicit differentiation problem

[PaulKraemer]

Hi,

I am stuck on the following implicit differentiation problem:

Suppose that x^3 + xy + y^4 = 19 determines a differentiable function f such that y = f(x). If P(1,2) is a point on the graph of f, use differentials to approximate the ordinate b of the point Q(1.10, b) on the graph.

I used implicit differentiation to find y ' and came up with:

y ' = (-3x^2 - y) / (x + 4y^3)

I then plugged the coordinates of P(1,2) into the above formula for y ' and got -5/33.

I calculated the differential as follows:

dy = y ' dx = (-5/33) * 0.10 = -0.015

I then added dy to the ordinate of P(1,2) to come up with b:

b = 2 - 0.015 = 1.85

The back of the book says the answer is b = 0.015.

I came up with my same answer twice. If anyone can tell me where I am going wrong, I'd really appreciate it.

Thanks in advance,
Paul

 

[galactus]

\(\displaystyle dy=\frac{-3x^{2}-y}{x+4y^{3}}dx\)

Entering in \(\displaystyle x=1, \;\ y=2, \;\ dx=.1\), gives \(\displaystyle b=\frac{1}{66}\approx .015\)

 

[PaulKraemer]

Hi Galactus,

When I plug x=1, y=2, and dx= .1 into...

dy = (-3x^2 - y) / (x + 4y^3) dx

...I get dy = -1/66 = -.015 (not 1/66 = .015 as listed in the back of my book and as you stated in your post. Am I missing something?)

...What also confuses me is that I don't think the problem asks for dy. It asks for the ordinate b of the point Q(1.10, b). To come up with b, wouldn't I have to add dy to the ordinate of P(1,2), which would be 2 - 0.015 = 1.985?

I apologize for my ignorance. I really appreciate your help.

Kind regards,
Paul

 

[galactus]

Yes, it should be -1/66. My booboo.

I also made a mistake by finding dy. You are correct. They are asking for the y coordinate, not dy.

You are certainly not ignorant. You are on the ball. I am the one who floundered.

I think the back of the book may be incorrect. They give the differential, but ask for the y coordinate when x=1.1

Yes, 1.985 is correct for b.

Good work.