Problem 19 - Applied Calculus

Question

For a function \(f(x, y),\) we are given \(f(100,20)=2750\) and \(f_{x}(100,20)=4,\) and \(f_{y}(100,20)=7 .\) Estimate \(f(105,21)\)

Step-by-Step Solution

Answer

Using linear approximation, estimate that \(f(105,21) \approx 2777\).

Step 1: Understand the Problem

We need to estimate the value of the function \(f(x, y)\) at the point (105, 21) given the function value and its partial derivatives at the point (100, 20).

Step 2: Recall the Linear Approximation Formula

The linear approximation formula is given by: \( f(x, y) \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \). We will apply this formula with \((x_0, y_0) = (100, 20)\).

Step 3: Substitute the Given Values

Substitute \(f(100,20) = 2750\), \(f_x(100,20) = 4\), \(f_y(100,20) = 7\), \(x_0 = 100\), \(y_0 = 20\), \(x = 105\), and \(y = 21\) into the formula.

Step 4: Simplify the Expression

Substitute the values into the formula: \( f(105, 21) \approx 2750 + 4(105 - 100) + 7(21 - 20) \).

Step 5: Calculate the Result

Perform the arithmetic: \( f(105, 21) \approx 2750 + 4 \times 5 + 7 \times 1 = 2750 + 20 + 7 = 2777 \).

Key Concepts

linear approximation

Linear approximation is a method used in multivariable calculus to estimate the value of a function at a given point. It is particularly useful when dealing with functions of several variables. The idea is to approximate a function using a linear equation near a point where the function’s value and behavior are known.

The basic formula for a linear approximation of a function \( f(x, y) \) near a point \((x_0, y_0)\) is:

• \( f(x, y) \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \)

This equation uses the known value of the function at \((x_0, y_0)\), which is \( f(x_0, y_0) \). It also incorporates the partial derivatives \( f_x(x_0, y_0) \) and \( f_y(x_0, y_0) \), which measure how the function changes with respect to \( x \) and \( y \), respectively. By adding these products, we create a tangent plane, which aptly approximates the function in the vicinity of \((x_0, y_0)\).

Linear approximation is akin to a zoomed-in view of the function, offering a simplified model that can be quickly computed to predict function behavior near known points.

partial derivatives

Partial derivatives are fundamental in multivariable calculus. They measure the rate at which a function changes as one of the variables is varied while all other variables are held constant.

For a given function \( f(x, y) \), its partial derivative with respect to \( x \), denoted by \( f_x(x, y) \), represents how \( f \) changes as \( x \) changes, with \( y \) kept constant. Likewise, \( f_y(x, y) \) represents the change of \( f \) as \( y \) changes, with \( x \) unaltered. These derivatives give us insights into the slopes of the surface defined by \( f \) in respective directions.

In our exercise, we used the values \( f_x(100, 20) = 4 \) and \( f_y(100, 20) = 7 \) to estimate how much the function \( f \) changes as \( x \) and \( y \) increase from 100 to 105 and 20 to 21, respectively.

Partial derivatives are critical in many scientific and engineering applications, helping to model complex systems and predict outcomes based on variable changes.

estimating function values

Estimating function values is a common challenge in calculus, especially with multivariable functions where exact values might be hard to determine directly.

In our given problem, estimating \( f(105, 21) \) involved knowing the behavior of \( f \) at a nearby point \( (100, 20) \). We found this estimation by employing the linear approximation technique. By substituting known values into the linear formula:

• \( f(105, 21) \approx 2750 + 4(105 - 100) + 7(21 - 20) \)

We mathematically predict function behavior by plugging in this data, calculating to find:

• \( f(105, 21) \approx 2750 + 20 + 7 = 2777 \)

Estimating functions can save time and computational resources by avoiding complex direct evaluations. It is particularly useful in real-world applications where precise measurement data at every single point is not feasible.

This method provides a close approximation that is highly valuable for quick assessments in various fields.