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Calculus/Correct to my Questions (Implicit Differentiation)

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Expert: - 3/21/2009


Question
1. Use implicit differentiation to find the equation of the tangent line to the curve xy^3+xy=16 at the point (8,1). The equation of this tangent line can be written in the form y=mx=b where m is: ______ and where b is: ________

2. find the slope of the tangent line to the curve
-3x^2-4xy-2y^3=-12 at the point (-2,2) .

3. Find the slope of the tangent line to the curve (a lemniscate) 2(x^2+y^2)^2=25(x^2-y^2) at the point (3,-1).

4. If 3x^2+3x+xy=4 and y(4)=-14, find y'(4) by implicit differentiation.

5. If (x^2)/(49) + (y^2)/(16)=1 and y(2)=3.83326, find y'(2) by implicit differentiation.  

Answer
Now the correct answer to the first 4 I just sent
and then I was this corrected case.

What I will do this time is look at 5.
(x/7)² + (y/4)² = 1 looks like an ellipse.

At x=2, let's see what y is.
(2/7)² + (y/4)² = 1, so y is 4√(45/49) = 3.83325939 (to a few more deciamls).

To find y', it can be seen that 2x/49 + 2yy'/16 = 0.
Putting in the values for x and y gives us
4/49 + 0.479157424y'= 0, so y' = (4/49)/0.479157424,
so y' = -0.170367084.  Now rounding to five places,
lke what y(2) was given as, I get -0.17037.

I didn't actually type out these numbers -
they were imported from Excel, where I did the calculations.
It's good to always keep numbers to as many places as you can
until you get the answer.

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