Thank you for visiting 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. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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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