(a) If there is a scalar function f(x, y) such that ∇ f(x, y) = x² i + y² j, then
∂f/∂x = x²
∂f/∂y = y²
Integrating both of these equations with respect to x and y (respectively) gives
f(x, y) = 1/3 x³ + g(y)
f(x, y) = 1/3 y³ + h(x)
Differentiating with respect to the other variable gives
∂f/∂y = g'(y) = y²
∂f/∂x = h'(x) = x²
so it follows that
f(x, y) = 1/3 x³ + 1/3 y³ + C
for some constant C.
(b) By the gradient theorem,
Answer:
Exterme Value: .
The bird reaches its minimum height which is 4ft affter covering a horizontal distance of 2 ft.
Step-by-step explanation:
The given function is .
We add and subtract the square of half the coefficient of x.
.
The first three terms form a perfect quadratic trinomial.
.
This function is now written in the form , where (h,k) is the extreme value(vertex).
The extreme value or the vertex is .
Interpretation:
The bird reaches its minimum height which is 4ft affter covering a horizontal distance of 2 ft.
Answer: x = 15
Step-by-step explanation:
First, we will have to acknowledge that Angle M is a complementary angle (that means that it is 90 degrees) meaning that 2x° + 4x° will equal 90 degrees. So, let’s set up this equation.
90 = 2x + 4x
90 = 6x
=
15 = x
Therefore, the x is 15.