If (x+2y)dy/dx=2x-y, what is the value of d^2y/dx^2 at the point (3,0)?
A. -10/3
B. 0
C. 2
D. 10/3
E. Undefined
using implicit derivatives,
(x+2y)y'=2x-y
(1+2y')y' + (x+2y)y" = 2-y'
Now plug in the numbers:
y'(3) = 2, so
(1+4)(2) + (3)y" = 2-2
10+3y" = 0
y" = -10/3
Well, well, well, we have a lovely differential equation here! Let's put on our mathematical clown wig and solve this puzzle, shall we?
First, let's rearrange the equation a bit. We have (x+2y)dy/dx = 2x-y. If we divide both sides by (x+2y), we get dy/dx = (2x-y)/(x+2y).
To find d^2y/dx^2, we need to take the derivative of dy/dx. But we're given a point (3,0), so we can use it to simplify things. At (3,0), we have (3+2(0))d(0)/dx = 2(3)-(0), which simplifies to 0 = 6.
Now, let's find d^2y/dx^2. To do that, we take the derivative of (2x-y)/(x+2y). This requires some chain and quotient rule fun!
Using the chain rule, we have d^2y/dx^2 = [(d/dx)(2x-y)(x+2y) - (2x-y)(d/dx)(x+2y)] / (x+2y)^2.
Okay, let's simplify this expression a bit. We know that dy/dx = 0 at (3,0), so we can substitute that in. Our expression becomes [0 - (2(3)-(0))(d/dx)(x+2y)] / (x+2y)^2.
Again, we can simplify this further. (2(3)-(0)) = 6, and (x+2y)^2 at (3,0) is 3^2 = 9. Our expression is now [0 - 6(d/dx)(x+2y)] / 9.
At (3,0), d/dx(x+2y) = d/dx(3+2(0)) = 2. So our expression simplifies to [0 - 6(2)] / 9 = -12/9 = -4/3.
Finally, we reach the end of our clownish journey! The value of d^2y/dx^2 at the point (3,0) is -4/3.
But hey, remember that the clown's answer isn't always the right one! Make sure to double-check my math, just in case. Happy solving!
To find the value of d^2y/dx^2 at the point (3,0) using the given equation, we can follow these steps:
Step 1: Rearrange the equation.
(x + 2y)dy/dx = 2x - y
dy/dx = (2x - y)/(x + 2y)
Step 2: Differentiate both sides of the equation with respect to x to find the second derivative.
d²y/dx² = d/dx[(2x - y)/(x + 2y)]
Step 3: Apply the quotient rule:
d²y/dx² = [(d/dx(2x - y)(x + 2y)) - (2x - y)(d/dx(x + 2y))] / (x + 2y)²
Step 4: Simplify the equation:
d²y/dx² = [(2(x + 2y) - (2x - y)(1 + 2(dy/dx)))/(x + 2y)²]
Step 5: Substitute the point values (3,0) into the equation:
d²y/dx² = [(2(3) - (2(3) - 0)(1 + 2(dy/dx)))/(3 + 2(0))²]
= [6 - (6)(1 + 2(dy/dx))]/9
= [6 - 6 - 12(dy/dx)]/9
= [-12(dy/dx)]/9
= -12(dy/dx)/9
= -4(dy/dx)/3
Step 6: Substitute the value of dy/dx at the point (3,0):
d²y/dx² = -4(0)/3 = 0
Therefore, the value of d²y/dx² at the point (3,0) is 0.
The answer is B. 0.
To find the value of d^2y/dx^2 at the point (3,0), we need to differentiate the given equation with respect to x.
Given: (x + 2y) dy/dx = 2x - y
Differentiating both sides of the equation with respect to x using the product rule, we get:
d/dx[(x + 2y) dy/dx] = d/dx(2x - y)
Using the Leibniz notation, the left side of the equation can be written as:
d/dx[(x + 2y) dy/dx] = d/dx(x + 2y) * dy/dx + (x + 2y) d^2y/dx^2
Differentiating the right side of the equation gives:
d/dx(2x - y) = 2 - dy/dx
Replacing these derivatives back into the equation, we have:
d/dx(x + 2y) * dy/dx + (x + 2y) d^2y/dx^2 = 2 - dy/dx
Now, let's substitute the point (3,0) into the equation to find the value of d^2y/dx^2 at that point.
At (3,0), we have x = 3 and y = 0.
Plugging these values into the equation gives:
d/dx(3 + 2(0)) * dy/dx + (3 + 2(0)) d^2y/dx^2 = 2 - dy/dx
Simplifying, we get:
d/dx(3) * dy/dx + 3 d^2y/dx^2 = 2 - dy/dx
Since dy/dx is not given, we cannot determine its value at the point (3,0).
Therefore, the value of d^2y/dx^2 at the point (3,0) is undefined.
Hence, the correct answer is E. Undefined.