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Consider the following function:

\[ f(x, y) = 5x^{1/4}y^{3/4} \]

Suppose that \( x = 99.8 \) and \( y = 101.5 \). Estimate \( f \) without using a calculator.

(Hint: Use total differentiation.)

Answer :

The estimated value of f(x, y) when x = 99.8 and y = 101.5, without using a calculator, is approximately 0.

To estimate the value of the function f(x, y) = 5x^(1/4) y^(3/4) without using a calculator, we can utilize the concept of total differentiation.

Total differentiation allows us to approximate the change in a function based on small changes in its variables. In this case, we can estimate the change in f(x, y) when x and y change by small amounts dx and dy, respectively.

Let's start by calculating the partial derivatives of f(x, y) with respect to x and y:

∂f/∂x = (5/4) * x^(-3/4) * y^(3/4)

∂f/∂y = (15/4) * x^(1/4) * y^(-1/4)

Next, we can use the total differential formula:

df ≈ (∂f/∂x) * dx + (∂f/∂y) * dy

Now, let's substitute the given values x = 99.8 and y = 101.5 into the partial derivatives:

∂f/∂x ≈ (5/4) * (99.8)^(-3/4) * (101.5)^(3/4)

∂f/∂y ≈ (15/4) * (99.8)^(1/4) * (101.5)^(-1/4)

Since we are interested in estimating f(x, y) with the given values, we can set dx = 0 and dy = 0, as we want to measure the change at a specific point:

df ≈ (∂f/∂x) * dx + (∂f/∂y) * dy

≈ (∂f/∂x) * 0 + (∂f/∂y) * 0

≈ 0

Therefore, the estimated value of f(x, y) when x = 99.8 and y = 101.5, without using a calculator, is approximately 0.

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