Suppose y is defined implicitly as a function of x by x^2+Axy^2+By^3=1 where A and B are constants to be determined. Given that this curve passes through the point (3,2) and that its tangent at this point has slope -1, find A and B.
take the derivative:
2x + (Ax)(2y)dy/dx + Ay^2 + 3By^2 dy/dx = 0
.....
...
dy/dx = (-2x - Ay^2)/(2Axy + 3By^2)
at (3,2) dy/dx = -1
-1 = (-6 - 4A)/(12A + 12B)
..
4A + 6B = 3
also in original,
9 + 12A + 8B = 1
3A + 2B = -2
triple the last one, and subtract ....
5A=-9
A=-9/5
sub back in to get B = 17/10
(Was expecting "nicer" numbers. I might have made an arithmetic error, check my arithmetic)
To solve this problem, we need to use the concept of implicit differentiation and the given information.
Step 1: Implicit differentiation
Differentiate both sides of the given equation with respect to x to find the derivative of y with respect to x. Remember to use the product rule and the chain rule where necessary.
d/dx (x^2 + Axy^2 + By^3) = d/dx (1)
2x + A(2xy * dy/dx + y^2) + B(3y^2 * dy/dx) = 0
Simplify the equation:
2x + 2Axy * dy/dx + Ay^2 + 3By^2 * dy/dx = 0
Step 2: Solve for dy/dx
Factor out dy/dx terms:
(2Axy + 3By^2) * dy/dx = -2x - Ay^2
Now, divide both sides by (2Axy + 3By^2) to isolate dy/dx:
dy/dx = (-2x - Ay^2) / (2Axy + 3By^2)
Step 3: Apply the given conditions
We are given that the curve passes through the point (3,2) and that its tangent at this point has a slope of -1.
Substitute x = 3 and y = 2 into the equation to represent the point (3,2):
dy/dx = (-2(3) - A(2)^2) / (2A(3)(2) + 3B(2)^2)
Simplify the expression:
-5 - 4A / (12A + 12B)
We are also given that the slope at this point is -1, so equate dy/dx to -1:
-5 - 4A / (12A + 12B) = -1
Step 4: Solve for A and B
Multiply both sides by (12A + 12B) to eliminate the denominator:
-5(12A + 12B) - 4A = -12A - 12B
-60A - 60B - 4A = -12A - 12B
Combine like terms:
-64A - 60B = -12A - 12B
Add 12A and 12B to both sides:
-64A - 48A = -12B + 60B
-112A = 48B
Divide by -112:
A = -(3/7)B
Therefore, A is -(3/7) times B.
This is the result for the values of A and B that satisfy the conditions of the problem.