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Chapter 16: Problem 20

Use differentials to approximate the change in \(f\) if the independent variables change as indicated. $$ \begin{aligned} f(x, y)=x^{2}-2 x y+3 y &; \\ (1.2) \text { to }(1.03,1.99) \end{aligned} $$

Short Answer

Expert verified
The change in \( f \) is approximately 0.526.

Step by step solution

01

Finding the Partial Derivatives

First, we need to find the partial derivatives of the function \( f(x, y) = x^2 - 2xy + 3y \) with respect to \( x \) and \( y \). The partial derivative of \( f \) with respect to \( x \) is: \[ f_x = \frac{\partial}{\partial x}(x^2 - 2xy + 3y) = 2x - 2y \]The partial derivative of \( f \) with respect to \( y \) is: \[ f_y = \frac{\partial}{\partial y}(x^2 - 2xy + 3y) = -2x + 3 \]
02

Evaluating Partial Derivatives at Initial Point

Evaluate the partial derivatives \( f_x \) and \( f_y \) at the initial point \((x_0, y_0) = (1.2, 1)\): \[ f_x(1.2, 1) = 2(1.2) - 2(1) = 2.4 - 2 = 0.4 \] \[ f_y(1.2, 1) = -2(1.2) + 3 = -2.4 + 3 = 0.6 \]
03

Calculating Differential Changes

Find the changes in the variables, \( \Delta x \) and \( \Delta y \): \[ \Delta x = 1.03 - 1.2 = -0.17 \]\[ \Delta y = 1.99 - 1 = 0.99 \]
04

Approximate the Change in Function

Use the linear approximation to find the change in \( f \): \[ df \approx f_x \cdot \Delta x + f_y \cdot \Delta y \]Plug in the values obtained:\[ df \approx 0.4 \cdot (-0.17) + 0.6 \cdot 0.99 \]\[ df \approx -0.068 + 0.594 \]\[ df \approx 0.526 \]
05

Conclusion

The approximate change in \( f \), \( df \), as the variables change from \((1.2, 1)\) to \((1.03, 1.99)\) is approximately 0.526.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
In calculus, a partial derivative represents how a function changes as one of the variables is varied, while every other variable is held constant. This is a key concept when dealing with functions of multiple variables.
  • A partial derivative is akin to taking the ordinary derivative of a one-variable function, but it is focused on one variable while the others are treated as constants.
  • For the function given, \( f(x, y) = x^2 - 2xy + 3y \), we found two partial derivatives: \( f_x \) and \( f_y \).
The equations for these derivatives are:
  • \( f_x = 2x - 2y \) 鈥� showing how the function changes with respect to \( x \) at any given point \( y \).
  • \( f_y = -2x + 3 \) 鈥� displaying the change concerning \( y \) while holding \( x \) constant.
These derivatives help us understand the slope or velocity of the function's surface in any given direction, a foundational idea in multivariable calculus.
Linear Approximation
Linear approximation is a method used to estimate the value of a function based on its derivatives at a certain point. This simplification allows for quick calculations and is widely used when small changes in variables occur.
  • It's based on the notion that you can represent a small segment of a potentially complex curve by a straight line tangent to the curve.
  • The formula used for linear approximation in this exercise is \( df \approx f_x \cdot \Delta x + f_y \cdot \Delta y \).
In our exercise:
  • The change in \( x \), \( \Delta x = 1.03 - 1.2 = -0.17 \), slightly shifts its value.
  • The change in \( y \), \( \Delta y = 1.99 - 1 = 0.99 \), provides another small movement.
  • Plug these into the formula to estimate the change in the function, \( f \), and you find the approximate change is \( df \approx 0.526 \).
This approach is very powerful for models that only require approximate solutions, especially in engineering and physics problems.
Function of Two Variables
A function of two variables relies on two separate inputs to generate an output. This type of function is the cornerstone of analyzing relationships where decisions are influenced by more than a single factor.
  • The function \( f(x, y) = x^2 - 2xy + 3y \) demonstrates a relationship dependent on both \( x \) and \( y \).
  • Each pair of \( (x, y) \) coordinates corresponds to a unique output \( f(x, y) \), visualized as a surface in 3D space.
Understanding these functions is crucial to deciphering how changes in variables affect outcomes, making them essential for fields like economics, engineering, and mathematics. In our particular case, moving from \((1.2, 1)\) to \((1.03, 1.99)\) reflects real-world scenarios where independent variables shift, requiring effective means of approximation and prediction.

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