For the following pair of functions f and g, determine if the level curves of the functions cross at right angles, and find their gradients at the point (1, 4).
f(x, y) = 5x + 5y , g(x, y) = 5x − 5y ;
To determine if the level curves of the functions f and g cross at right angles, we need to check if their gradients are orthogonal at any given point.
Let's start by finding the gradients of the functions f and g.
The gradient (or the partial derivatives) of a function f(x, y) with respect to x is denoted as ∂f/∂x, and the gradient with respect to y is denoted as ∂f/∂y.
For the function f(x, y) = 5x + 5y, the partial derivatives are:
∂f/∂x = 5
∂f/∂y = 5
Similarly, for the function g(x, y) = 5x - 5y, the partial derivatives are:
∂g/∂x = 5
∂g/∂y = -5
Now, let's find the gradients at the given point (1, 4).
The gradients at (1, 4) for f and g are:
∇f = (∂f/∂x, ∂f/∂y) = (5, 5)
∇g = (∂g/∂x, ∂g/∂y) = (5, -5)
To determine if the gradients are orthogonal or perpendicular, we can take the dot product of the gradients.
∇f · ∇g = (5)(5) + (5)(-5) = 25 + (-25) = 0
Since the dot product is zero, it means that the gradients of f and g at (1, 4) are orthogonal, indicating that their level curves cross at right angles.
Therefore, the level curves of the functions f(x, y) = 5x + 5y and g(x, y) = 5x - 5y cross at right angles, and the gradients at the point (1, 4) are ∇f = (5, 5) and ∇g = (5, -5).