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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