Given a curve, say y^3 = x^2, and a value, say 4.
How can I find a point in which the curve will have slope 4?
I understand the equation for slope in this case would be: dy/dx = 2x/3y. But how can you find a point that satisfies both 2x/3y and y^3 = x^2?
for your given example , dy/dx = 2x/(3y^2)
so you set that equal to 4
2x/(3y^2) = 4
2x = 12y^2
x = 6y^2
now plug that back into the original y^3 = x^2
y^3 = 36y^4
36y^4 - y^3 = 0
y^3(36y - 1) = 0
y = 0 or y = 1/36
if y = 0, the x = 0 <------ ****
if y = 1/36, then x = 6(1/36)^2 = 1/216
However, the derivative is indeterminate at (0,0), it would be 0/0
take a point very very close to the origin.
let y = .001
then x^2 = .001^3 , using my calculator ....
x = appr .000031622 and
dy/dx = 2x/(3y^2) = appr 666,667 appears to be approaching infinitiy
for the other point (1/216,1/36)
dy/dx = (1/108) / (3(1/1296)
= (1/108)(1296/3) = 4 , as needed.
To find a point on the curve y^3 = x^2 where the curve has a slope of 4, you can use calculus to compute the derivative of the equation with respect to x. The derivative will give you the slope of the curve at any point.
1. Start with the equation y^3 = x^2.
2. Take the derivative of both sides of the equation with respect to x. The derivative of y^3 with respect to x can be found using the chain rule, while the derivative of x^2 is simply 2x.
d/dx(y^3) = d/dx(x^2)
3y^2 * dy/dx = 2x
3. Rearrange the equation to solve for dy/dx, which represents the slope of the curve.
dy/dx = 2x / (3y^2)
4. Plug in the slope value of 4 into the equation:
4 = 2x / (3y^2)
5. Now, substitute the given value of x, which is 4, into the equation:
4 = 2(4) / (3y^2)
4 = 8 / (3y^2)
6. Solve the equation for y:
Cross-multiply: 4 * (3y^2) = 8
12y^2 = 8
Divide both sides by 12: y^2 = 8/12
Simplify: y^2 = 2/3
Take the square root of both sides to find y: y = ±√(2/3)
7. Substitute the values of x and y back into the original equation to get the point(s) where the slope is 4:
For y = √(2/3):
(x, y) = (4, √(2/3))
For y = -√(2/3):
(x, y) = (4, -√(2/3))
So, the points on the curve y^3 = x^2 where the slope is 4 are (4, √(2/3)) and (4, -√(2/3)).