Final answer:
To develop a one-dimensional function in the temperature gradient direction at the point (1, 1) using the given temperature function, we need to find the gradient vector of the temperature function at that point.
Step-by-step explanation:
To develop a one-dimensional function in the temperature gradient direction at the point (1, 1) using the given temperature function, we need to find the gradient vector of the temperature function at that point. The gradient vector represents the direction and rate of change of the function. The gradient vector is obtained by taking the partial derivatives of the temperature function with respect to each variable, x and y, and evaluating them at the given point.
Let's calculate the gradient vector step by step:
Step 1: Calculate the partial derivative of the temperature function with respect to x:
f'(x) = 6xy^2 - 7y - 2x
Step 2: Calculate the partial derivative of the temperature function with respect to y:
f'(y) = 4x^3y - 7x + 3
Step 3: Evaluate the partial derivatives at the given point (1, 1):
f'(1) = 6(1)(1)^2 - 7(1) - 2(1) = -3
f'(1) = 4(1)^3(1) - 7(1) + 3 = 0
Step 4: The gradient vector at the point (1, 1) is:
∇f(1, 1) = (-3, 0)
This gradient vector represents the direction and rate of change of the temperature function at the point (1, 1). Therefore, a one-dimensional function in the temperature gradient direction at the point (1, 1) can be written as:
f'(t) = -3t
where t represents the distance along the gradient vector.