the point (1,9) lies on the graph of an equation y=f(x) for which dy/dx=4x(sqrt(y)) where x is greater than or equal to 0 and y is greater than or equal to 0
a)6
b)4
c)2
d)sqrt(2)
e)0
dy/dx = 4x*y^1/2
y^(-1/2) dy = 4x dx
2y^1/2 = 2x^2 + c
y^1/2 = x^2 + C
3 = 1 + C
C = 2
√y = x^2 + 2
now, what's your question?
To find the value of f(x) at the point (1,9), we can use the given differential equation and integrate it.
dy/dx = 4x(sqrt(y))
Separating variables, we get:
1/sqrt(y) dy = 4x dx
Integrating both sides:
∫(1/sqrt(y)) dy = ∫4x dx
∫y^(-1/2) dy = 4∫x dx
Applying the power rule of integration, we have:
2√y = 2x^2 + C
Where C is the constant of integration.
To find the value of C, we can use the fact that the point (1,9) lies on the graph. Plugging in the values x = 1 and y = 9 into the equation:
2√(9) = 2(1)^2 + C
6 = 2 + C
C = 6 - 2 = 4
Plugging C back into the equation, we have:
2√y = 2x^2 + 4
Dividing both sides by 2:
√y = x^2 + 2
Squaring both sides:
y = (x^2 + 2)^2
Therefore, the equation of the graph is y = (x^2 + 2)^2.
Now, let's find f(1). Plugging x = 1 into the equation:
f(1) = (1^2 + 2)^2
f(1) = (1 + 2)^2
f(1) = 3^2
f(1) = 9
So, the value of f(x) at the point (1,9) is 9.
Therefore, the answer is (a) 6.
To find the value of f'(1), we can use the given equation and the point (1, 9).
1. Start with the equation: dy/dx = 4x(sqrt(y))
2. Substitute x = 1 and y = 9 into the equation:
dy/dx = 4(1)(sqrt(9))
dy/dx = 4(3)
dy/dx = 12
So, f'(1) = 12.
Therefore, the answer is not one of the options provided (a) 6, b) 4, c) 2, d) sqrt(2), e) 0).