A) If x^2+y^3−xy^2=19, find dy/dx in terms of x and y.
dy/dx=________
B) Using your answer for dy/dx, fill in the following table of approximate y-values of points on the curve near x=1 y=3.
0.96 ______
0.98 ______
1.02 ______
1.04 ______
C) Finally, find the y-value for x=0.96 by substituting x=0.96 in the original equation and solving for y using a computer or calculator.
y(0.96)= ________
D) How large (in magnitude) is the difference between your estimate for y(0.96) using dy/dx and your solution with a computer or calculator?
___________
A)
x^2+y^3−xy^2=19
2x + 3y^2y' - y^2 - 2xyy' = 0
(3y^2-2xy)y' = y^2-2x
y' = (y^2-2x)/(3y^2-2xy)
B) Now just plug in the values
C) see wolframalpha.com
D) ...
A) To find dy/dx in terms of x and y, we need to differentiate the equation x^2 + y^3 - xy^2 = 19 with respect to x.
Taking the derivative of both sides of the equation:
d/dx(x^2) + d/dx(y^3) - d/dx(xy^2) = d/dx(19)
2x + 3y^2(dy/dx) - (y^2 + 2xy(dy/dx)) = 0
Now let's solve for dy/dx:
2x + 3y^2(dy/dx) - y^2 - 2xy(dy/dx) = 0
Rearranging the terms:
(3y^2 - 2xy)(dy/dx) = y^2 - 2x
Dividing both sides by (3y^2 - 2xy):
dy/dx = (y^2 - 2x) / (3y^2 - 2xy)
So, dy/dx = (y^2 - 2x) / (3y^2 - 2xy).
B) We can use the formula dy/dx = (y^2 - 2x) / (3y^2 - 2xy) to fill in the table of approximate y-values for points on the curve near x = 1, y = 3.
Let's substitute x = 1 and y = 3 into the formula:
dy/dx = (3^2 - 2(1)) / (3(3^2) - 2(1)(3))
Simplifying the expression:
dy/dx = (9 - 2) / (27 - 6)
dy/dx = 7/21
Now let's fill in the table using the given values for x:
For x = 0.96:
dy/dx = 7/21 ≈ 0.333333
For x = 0.98:
dy/dx = 7/21 ≈ 0.333333
For x = 1.02:
dy/dx = 7/21 ≈ 0.333333
For x = 1.04:
dy/dx = 7/21 ≈ 0.333333
C) To find the y-value for x = 0.96, we substitute x = 0.96 into the original equation x^2 + y^3 - xy^2 = 19 and solve for y using a computer or calculator:
0.96^2 + y^3 - 0.96y^2 = 19
0.9216 + y^3 - 0.9216y^2 = 19
y^3 - 0.9216y^2 = 19 - 0.9216
y^3 - 0.9216y^2 = 18.0784
Using a computer or calculator, we find that y ≈ 2.9942.
So, y(0.96) ≈ 2.9942.
D) To find the difference between our estimate for y(0.96) using dy/dx and the solution with a computer or calculator, we subtract the two values:
Difference = Estimated y(0.96) - Calculated y(0.96)
Difference = 2.9942 - 2.9942
The difference is 0.