Asked by Anonymous
Consider a curve given implicitly by the equation (1+x)y^3 + (x^4)y - 85 = 0.
A. Calculate dy/dx at a general point (x,y).
B. Write the equation of the tangent line to the curve at the point (3,1).
C. At (3,1), y(x) is defined implicitly as a function of x. Let g(x) be the inverse function of y(x). Compute g'(1).
A. Calculate dy/dx at a general point (x,y).
B. Write the equation of the tangent line to the curve at the point (3,1).
C. At (3,1), y(x) is defined implicitly as a function of x. Let g(x) be the inverse function of y(x). Compute g'(1).
Answers
Answered by
Steve
A
(1+x)y^3 + (x^4)y - 85 = 0
y^3 + 3(1+x)y^2y' + 4x^3y + x^4y' = 0
y' = -(y^3+4x^3y)/(3(1+x)+x^4)
= -(y^3+4x^3y)/(x^4+3x+3)
B
y'(3) = -109/93
So, the tangent line is
y-1 = -109/93 (x-3)
C
y(3) = 1, so g(1) = 3
g'(1) = 1/y'(3) = -93/109
(1+x)y^3 + (x^4)y - 85 = 0
y^3 + 3(1+x)y^2y' + 4x^3y + x^4y' = 0
y' = -(y^3+4x^3y)/(3(1+x)+x^4)
= -(y^3+4x^3y)/(x^4+3x+3)
B
y'(3) = -109/93
So, the tangent line is
y-1 = -109/93 (x-3)
C
y(3) = 1, so g(1) = 3
g'(1) = 1/y'(3) = -93/109
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.