Question

suppose f is a differentiable function of x and y, and g(r, s) = f(5r − s, s2 − 7r). use the table of values below to calculate gr(4, 2) and gs(4, 2). f g fx fy (18, −24) 2 3 7 9 (4, 2) 3 2 6 5

Answer 1

The partial derivatives of the function g(r, s) at points (4, 2) can be found using the chain rule and the derivatives of f alongside the partial derivatives of x and y with respect to r and s.

Step-by-step explanation:

The problem asks us to calculate the partial derivatives of the function g(r, s) at the point (4, 2), given that g(r, s) is defined in terms of another function f(x, y) where x = 5r - s and y = s^2 - 7r.

Using the chain rule for partial derivatives, gr and gs can be computed through a combination of the provided derivatives fx and fy at given points, and the partial derivatives of x and y with respect to r and s. A table of values is given for function f and its derivatives which can be used to find the required derivatives of g at (4, 2).

To find gr(4, 2), we use the chain rule:

gr = fx xr + fy yr

Similarly, gs(4, 2) is:

gs = fx xs + fy ys

By substituting the values of fx and fy at the point (4, 2), and calculating the partial derivatives xr, xs, yr, ys, the desired values of gr and gs can be obtained.

The complete question is:content loaded

suppose f is a differentiable function of x and y, and g(r, s) = f(5r − s, s2 − 7r). use the table of values below to calculate gr(4, 2) and gs(4, 2). f g fx fy (18, −24) 2 3 7 9 (4, 2) 3 2 6 5 is:

Answer 2

The value of gs(4, 2) = 5 * 2 = 10.

gr(4, 2) = 30 and gs(4, 2) = 10.

Step-by-step explanation:

To calculate gr(4, 2) and gs(4, 2) using the table of values provided, we need to use the chain rule of differentiation.

Let's start with gr(4, 2):

gr(4, 2) represents the partial derivative of g with respect to r at the point (4, 2).

Using the chain rule, we have:

gr(4, 2) = (d/dx) [f(5x - y,- 7x)] * (d/dx) [5r - s]

The first part, (d/dx) [f(5x - y, - 7x)], represents the partial derivative of f with respect to x, evaluated at (5r - s, - 7r).

Looking at the given table, we can see that fx = 6 at the point (4, 2). Therefore, (d/dx) [f(5x - y, - 7x)] = 6.

The second part, (d/dx) [5r - s], represents the partial derivative of 5r - s with respect to r.

Taking the derivative with respect to r, we get:

(d/dx) [5r - s] = 5.

Therefore, gr(4, 2) = 6 * 5 = 30.

Now, let's calculate gs(4, 2):

gs(4, 2) represents the partial derivative of g with respect to s at the point (4, 2).

Using the chain rule, we have:

gs(4, 2) = (d/dx) [f(5x - y, - 7x)] * (d/dx) [ - 7r]

Using the given table, fy = 5 at the point (4, 2). Therefore, (d/dx) [f(5x - y, - 7x)] = 5.

Taking the derivative of - 7r with respect to s, we get:

(d/dx) - 7r] = 2s.