Question

Suppose the temperature of a flat surface at any point (x, y) is described by the function T (x, y) = ln(e^x + y^2).

(a) Determine the rate of change in T at the point (0, 0) in the direction of (−1, −2).
(b) In what direction does T increase most rapidly from point (0, 1)? What is the value of this rate of change?

Answer 1

The rate of change of T at the point (0, 0) in the direction of (-1, -2) is -1. The direction in which T increases most rapidly from the point (0, 1) can be determined by finding the gradient vector and normalizing it. The rate of change of T in that direction can be calculated by finding the magnitude of the gradient vector at the point (0, 1).

Step-by-step explanation:

(a) To determine the rate of change of T at the point (0, 0) in the direction of (-1, -2), we need to find the gradient of T at that point. The gradient represents the rate of change of T with respect to x and y. We can find the partial derivatives of T with respect to x and y and substitute the values of x and y as 0 in the resulting equations. The partial derivative with respect to x is obtained by differentiating T with respect to x while treating y as a constant. Similarly, the partial derivative with respect to y is obtained by differentiating T with respect to y while treating x as a constant. By substituting x = 0 and y = 0 in the partial derivatives, we get the rate of change at the point (0, 0) in the direction of (-1, -2) as -1.

(b) To determine the direction in which T increases most rapidly from the point (0, 1), we need to find the gradient of T at that point and then find the direction in which the gradient vector points. The gradient vector points in the direction of the maximum rate of change of T. Mathematically, the direction of the gradient vector is given by the unit vector obtained by normalizing the gradient vector. The magnitude of the gradient vector gives us the rate of change of T in that direction. By finding the partial derivatives of T with respect to x and y, substituting x = 0 and y = 1, normalizing the gradient vector, and calculating the magnitude, we can determine the direction and value of the rate of change at the point (0, 1).

Answer 2

The rate of change in T at the point (0, 0) in the direction of (-1, -2) is -1. The direction in which T increases most rapidly from point (0, 1) is (-1/2, -1), and the rate of change in T from point (0, 1) in this direction is √(5)/2.

Step-by-step explanation:

To determine the rate of change in T at the point (0, 0) in the direction of (-1, -2), we can use the gradient vector. The gradient vector is given by (partial derivative of T with respect to x, partial derivative of T with respect to y).

Taking the partial derivatives of T(x, y) = ln(eˣ + y²), we get:

a. Tx(x, y) = 1 / (eˣ + y²)

Ty(x, y) = 2y / (eˣ + y²)

Substituting the values for x and y in the gradient vector:

Gradient vector = (Tx(0, 0), Ty(0, 0)) = (1 / 1, 0) = (1, 0)

The rate of change in T at the point (0, 0) in the direction of (-1, -2) is given by the dot product of the gradient vector and the direction vector:

Rate of change = Gradient vector dot Direction vector

Rate of change = (1, 0) dot (-1, -2) = 1 × (-1) + 0 × (-2) = -1

Therefore, the rate of change in T at the point (0, 0) in the direction of (-1, -2) is -1.

b. To determine the direction in which T increases most rapidly from point (0, 1), we can again use the gradient vector. Evaluating the gradient vector at the point (0, 1):

Gradient vector = (Tx(0, 1), Ty(0, 1))

Gradient vector = (1 / (e⁰ + 1²), 2(1) / (e⁰ + 1²))

Gradient vector = (1/2, 1)

The direction opposite to the gradient vector is the direction in which T increases most rapidly. So the direction is (-1/2, -1). The magnitude (length) of the gradient vector represents the rate of change.

Therefore, the rate of change in T from point (0, 1) in the direction (-1/2, -1) is the magnitude of the gradient vector:

Rate of change = |Gradient vector| = |(1/2, 1)| = √((1/2)² + 1²) = √(1/4 + 1) = √(5/4) = √(5)/2

Therefore, the rate of change in T from point (0, 1) in the direction (-1/2, -1) is √(5)/2.