Calculus
posted by Mathew .
Consider the curve given by y^2 = 2+xy
(a) show that dy/dx= y/(2yx)
(b) Find all points (x,y) on the curve where the line tangent to the curve has slope 1/2.
(c) Show that there are now points (x,y) on the curve where the line tangent to the curve is horizontal.
(d) Let x and y be functions of time t that are related by the equation y^2 = 2 + xy. At time t=5, the value of y is 3 and dy/dt=6. Find the value of dx/dt at time t=5.
I was able to do part A, that was easy, but then with the rest of the parts I didn't know where to start. I know I should plug in 1/2 for the slope and solve but I have two variables of x and y. How should I go about solving these problems?

(B) You have two simultaneous equations in two variables. The original equation, and the derivative for specified values of dy/dx.
When dy/dx=1/2 :
1/2 = y/(2yx)
Rearrange:
2y  x = 2y
Hence: x = 0
Substitute back into y^{2} = 2 + xy:
Therefore: y = ±√2
Answer: (0, √2), and (0, +√2)
Respond to this Question
Similar Questions

last calc question, i promise!
given the curve x + xy + 2y^2 = 6... a. find an expression for the slope of the curve. i got (1y)/(x + 4y) as my answer. b. write an equation for the line tangent to the curve at the point (2,1). i got y = (1/3)x + (5/3). but i … 
12th Grade Calculus
1. a.) Find an equation for the line perpendicular to the tangent curve y=x^3  9x + 5 at the point (3,5) [* for a. the answer that I obtained was y5 = 1/18 (x3) ] b.) What is the smallest slope on the curve? 
Calculus
Consider the curve y^2+xy+x^2=15. What is dy/dx? 
calculus
1. Given the curve a. Find an expression for the slope of the curve at any point (x, y) on the curve. b. Write an equation for the line tangent to the curve at the point (2, 1) c. Find the coordinates of all other points on this curve … 
calculus
Consider the curve defined by 2y^3+6X^2(y) 12x^2 +6y=1 . a. Show that dy/dx= (4x2xy)/(x^2+y^2+1) b. Write an equation of each horizontal tangent line to the curve. c. The line through the origin with slope 1 is tangent to the curve … 
Calculus
Please help this is due tomorrow and I don't know how to Ive missed a lot of school sick Consider the curve given by the equation x^3+3xy^2+y^3=1 a.Find dy/dx b. Write an equation for the tangent line to the curve when x = 0. c. Write … 
calculus
Given the curve defined by the equation y=cos^2(x) + sqrt(2)* sin(x) with domain (0,pi) , find all points on the curve where the tangent line to the curve is horizontal 
Math
In this problem we consider drawing some straight lines which form a nice pattern. Consider joining the point (0.1,0) to the point (0,0.9) by a line segment; then joining (0.2,0) to (0,0.8) by a line segment; and so on. In general, … 
Calculus
Can someone show how this question is solved. Consider the curve given by the equation 2y^3 + y^2  y^5 = x^4 2x^3 +x^2 Find all points at which the tangent line to the curve is horizontal or vertical. Thanks! 
Calculus AB
Could someone please help me with these tangent line problems?