Question

The equation dy/dx = -6x^2/y gives the slope at any point on the graph of f(x). The range of f(x) is [0, infinity] and f(1) = 2.

A. Find the equation of the tangent line to f(x) at the point (1,2).

B. Write the function f(x).

C. Determine the domain of the function f(x).

Answer 1

we know the slope is dy/dx, so at (1,2), the slope is -3.

The tangent line is thus

y-2 = -3(x-1)



y dy = -6x^2 dx
1/2 y^2 = -2x^3 + c
y = √2 √(c-2x^3)

you know that the domain of √ is non-negative numbers, so

c-2x^3 >= 0
2x^3 <= c
x <= ∛(c/2)

Answer 2

To compound off of what Steve said, because the point (1,2) is known, you can plug it into y to get C to be equal to 8. Thus, the domain of the function of would be [-∞, ∛2]. Let me know if you want to see where I got these numbers from